Arithmetic Sequence — Formula for the General Sum — Dynamic Practice
Arithmetic Sequence — Formula for the General Sum — Dynamic Practice. Practice questions to deepen understanding of finding the formula for the sum of the first n terms in an arithmetic sequence. Online math practice with full solutions and detailed explanations.
Dynamic practice in deriving the formula for Sₙ — using Sₙ = n(2a₁ + (n−1)d)/2 from given information. New questions every attempt.
Question 1
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 3\)
• The common difference: \(d = 16\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 3\)
• The common difference: \(d = 16\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 3 + (n-1) \cdot 16)}{2}\)
\(S_n = \frac{6n + 16n^2 - 16n}{2}\)
\(S_n = 8n² - 5n\)
\(S_n = \frac{6n + 16n^2 - 16n}{2}\)
\(S_n = 8n² - 5n\)
Answer: 8n² - 5n
Question 2
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 14\)
• The common difference: \(d = 10\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 14\)
• The common difference: \(d = 10\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 14 + (n-1) \cdot 10)}{2}\)
\(S_n = \frac{28n + 10n^2 - 10n}{2}\)
\(S_n = 5n² + 9n\)
\(S_n = \frac{28n + 10n^2 - 10n}{2}\)
\(S_n = 5n² + 9n\)
Answer: 5n² + 9n
Question 3
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 2\)
• The common difference: \(d = 4\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 2\)
• The common difference: \(d = 4\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 2 + (n-1) \cdot 4)}{2}\)
\(S_n = \frac{4n + 4n^2 - 4n}{2}\)
\(S_n = 2n²\)
\(S_n = \frac{4n + 4n^2 - 4n}{2}\)
\(S_n = 2n²\)
Answer: 2n²
Question 4
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 14\)
• The common difference: \(d = 2\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 14\)
• The common difference: \(d = 2\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 14 + (n-1) \cdot 2)}{2}\)
\(S_n = \frac{28n + 2n^2 - 2n}{2}\)
\(S_n = 1n² + 13n\)
\(S_n = \frac{28n + 2n^2 - 2n}{2}\)
\(S_n = 1n² + 13n\)
Answer: 1n² + 13n
Question 5
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 16\)
• The common difference: \(d = 14\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 16\)
• The common difference: \(d = 14\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 16 + (n-1) \cdot 14)}{2}\)
\(S_n = \frac{32n + 14n^2 - 14n}{2}\)
\(S_n = 7n² + 9n\)
\(S_n = \frac{32n + 14n^2 - 14n}{2}\)
\(S_n = 7n² + 9n\)
Answer: 7n² + 9n
Question 6
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 1\)
• The common difference: \(d = 4\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 1\)
• The common difference: \(d = 4\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 1 + (n-1) \cdot 4)}{2}\)
\(S_n = \frac{2n + 4n^2 - 4n}{2}\)
\(S_n = 2n² - 1n\)
\(S_n = \frac{2n + 4n^2 - 4n}{2}\)
\(S_n = 2n² - 1n\)
Answer: 2n² - 1n
Question 7
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 7\)
• The common difference: \(d = 8\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 7\)
• The common difference: \(d = 8\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 7 + (n-1) \cdot 8)}{2}\)
\(S_n = \frac{14n + 8n^2 - 8n}{2}\)
\(S_n = 4n² + 3n\)
\(S_n = \frac{14n + 8n^2 - 8n}{2}\)
\(S_n = 4n² + 3n\)
Answer: 4n² + 3n
Question 8
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 12\)
• The common difference: \(d = 12\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 12\)
• The common difference: \(d = 12\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 12 + (n-1) \cdot 12)}{2}\)
\(S_n = \frac{24n + 12n^2 - 12n}{2}\)
\(S_n = 6n² + 6n\)
\(S_n = \frac{24n + 12n^2 - 12n}{2}\)
\(S_n = 6n² + 6n\)
Answer: 6n² + 6n
Question 9
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 17\)
• The common difference: \(d = 4\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 17\)
• The common difference: \(d = 4\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 17 + (n-1) \cdot 4)}{2}\)
\(S_n = \frac{34n + 4n^2 - 4n}{2}\)
\(S_n = 2n² + 15n\)
\(S_n = \frac{34n + 4n^2 - 4n}{2}\)
\(S_n = 2n² + 15n\)
Answer: 2n² + 15n
Question 10
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 5\)
• The common difference: \(d = 12\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 5\)
• The common difference: \(d = 12\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 5 + (n-1) \cdot 12)}{2}\)
\(S_n = \frac{10n + 12n^2 - 12n}{2}\)
\(S_n = 6n² - 1n\)
\(S_n = \frac{10n + 12n^2 - 12n}{2}\)
\(S_n = 6n² - 1n\)
Answer: 6n² - 1n
Question 11
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 18\)
• The common difference: \(d = 20\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 18\)
• The common difference: \(d = 20\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 18 + (n-1) \cdot 20)}{2}\)
\(S_n = \frac{36n + 20n^2 - 20n}{2}\)
\(S_n = 10n² + 8n\)
\(S_n = \frac{36n + 20n^2 - 20n}{2}\)
\(S_n = 10n² + 8n\)
Answer: 10n² + 8n
Question 12
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 5\)
• The common difference: \(d = 4\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 5\)
• The common difference: \(d = 4\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 5 + (n-1) \cdot 4)}{2}\)
\(S_n = \frac{10n + 4n^2 - 4n}{2}\)
\(S_n = 2n² + 3n\)
\(S_n = \frac{10n + 4n^2 - 4n}{2}\)
\(S_n = 2n² + 3n\)
Answer: 2n² + 3n
Question 13
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 1\)
• The common difference: \(d = 20\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 1\)
• The common difference: \(d = 20\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 1 + (n-1) \cdot 20)}{2}\)
\(S_n = \frac{2n + 20n^2 - 20n}{2}\)
\(S_n = 10n² - 9n\)
\(S_n = \frac{2n + 20n^2 - 20n}{2}\)
\(S_n = 10n² - 9n\)
Answer: 10n² - 9n
Question 14
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 6\)
• The common difference: \(d = 18\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 6\)
• The common difference: \(d = 18\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 6 + (n-1) \cdot 18)}{2}\)
\(S_n = \frac{12n + 18n^2 - 18n}{2}\)
\(S_n = 9n² - 3n\)
\(S_n = \frac{12n + 18n^2 - 18n}{2}\)
\(S_n = 9n² - 3n\)
Answer: 9n² - 3n
Question 15
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 10\)
• The common difference: \(d = 4\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 10\)
• The common difference: \(d = 4\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 10 + (n-1) \cdot 4)}{2}\)
\(S_n = \frac{20n + 4n^2 - 4n}{2}\)
\(S_n = 2n² + 8n\)
\(S_n = \frac{20n + 4n^2 - 4n}{2}\)
\(S_n = 2n² + 8n\)
Answer: 2n² + 8n
Question 16
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 8\)
• The common difference: \(d = 4\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 8\)
• The common difference: \(d = 4\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 8 + (n-1) \cdot 4)}{2}\)
\(S_n = \frac{16n + 4n^2 - 4n}{2}\)
\(S_n = 2n² + 6n\)
\(S_n = \frac{16n + 4n^2 - 4n}{2}\)
\(S_n = 2n² + 6n\)
Answer: 2n² + 6n
Question 17
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 3\)
• The common difference: \(d = 2\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 3\)
• The common difference: \(d = 2\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 3 + (n-1) \cdot 2)}{2}\)
\(S_n = \frac{6n + 2n^2 - 2n}{2}\)
\(S_n = 1n² + 2n\)
\(S_n = \frac{6n + 2n^2 - 2n}{2}\)
\(S_n = 1n² + 2n\)
Answer: 1n² + 2n
Question 18
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 14\)
• The common difference: \(d = 6\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 14\)
• The common difference: \(d = 6\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 14 + (n-1) \cdot 6)}{2}\)
\(S_n = \frac{28n + 6n^2 - 6n}{2}\)
\(S_n = 3n² + 11n\)
\(S_n = \frac{28n + 6n^2 - 6n}{2}\)
\(S_n = 3n² + 11n\)
Answer: 3n² + 11n
Question 19
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 2\)
• The common difference: \(d = 16\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 2\)
• The common difference: \(d = 16\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 2 + (n-1) \cdot 16)}{2}\)
\(S_n = \frac{4n + 16n^2 - 16n}{2}\)
\(S_n = 8n² - 6n\)
\(S_n = \frac{4n + 16n^2 - 16n}{2}\)
\(S_n = 8n² - 6n\)
Answer: 8n² - 6n
Question 20
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 20\)
• The common difference: \(d = 10\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 20\)
• The common difference: \(d = 10\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 20 + (n-1) \cdot 10)}{2}\)
\(S_n = \frac{40n + 10n^2 - 10n}{2}\)
\(S_n = 5n² + 15n\)
\(S_n = \frac{40n + 10n^2 - 10n}{2}\)
\(S_n = 5n² + 15n\)
Answer: 5n² + 15n
Question 21
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 7\)
• The common difference: \(d = 20\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 7\)
• The common difference: \(d = 20\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 7 + (n-1) \cdot 20)}{2}\)
\(S_n = \frac{14n + 20n^2 - 20n}{2}\)
\(S_n = 10n² - 3n\)
\(S_n = \frac{14n + 20n^2 - 20n}{2}\)
\(S_n = 10n² - 3n\)
Answer: 10n² - 3n
Question 22
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 11\)
• The common difference: \(d = 2\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 11\)
• The common difference: \(d = 2\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 11 + (n-1) \cdot 2)}{2}\)
\(S_n = \frac{22n + 2n^2 - 2n}{2}\)
\(S_n = 1n² + 10n\)
\(S_n = \frac{22n + 2n^2 - 2n}{2}\)
\(S_n = 1n² + 10n\)
Answer: 1n² + 10n
Question 23
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 12\)
• The common difference: \(d = 20\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 12\)
• The common difference: \(d = 20\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 12 + (n-1) \cdot 20)}{2}\)
\(S_n = \frac{24n + 20n^2 - 20n}{2}\)
\(S_n = 10n² + 2n\)
\(S_n = \frac{24n + 20n^2 - 20n}{2}\)
\(S_n = 10n² + 2n\)
Answer: 10n² + 2n
Question 24
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 13\)
• The common difference: \(d = 6\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 13\)
• The common difference: \(d = 6\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 13 + (n-1) \cdot 6)}{2}\)
\(S_n = \frac{26n + 6n^2 - 6n}{2}\)
\(S_n = 3n² + 10n\)
\(S_n = \frac{26n + 6n^2 - 6n}{2}\)
\(S_n = 3n² + 10n\)
Answer: 3n² + 10n
Question 25
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 6\)
• The common difference: \(d = 6\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 6\)
• The common difference: \(d = 6\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 6 + (n-1) \cdot 6)}{2}\)
\(S_n = \frac{12n + 6n^2 - 6n}{2}\)
\(S_n = 3n² + 3n\)
\(S_n = \frac{12n + 6n^2 - 6n}{2}\)
\(S_n = 3n² + 3n\)
Answer: 3n² + 3n
Question 26
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 2\)
• The common difference: \(d = 18\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 2\)
• The common difference: \(d = 18\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 2 + (n-1) \cdot 18)}{2}\)
\(S_n = \frac{4n + 18n^2 - 18n}{2}\)
\(S_n = 9n² - 7n\)
\(S_n = \frac{4n + 18n^2 - 18n}{2}\)
\(S_n = 9n² - 7n\)
Answer: 9n² - 7n
Question 27
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 8\)
• The common difference: \(d = 2\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 8\)
• The common difference: \(d = 2\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 8 + (n-1) \cdot 2)}{2}\)
\(S_n = \frac{16n + 2n^2 - 2n}{2}\)
\(S_n = 1n² + 7n\)
\(S_n = \frac{16n + 2n^2 - 2n}{2}\)
\(S_n = 1n² + 7n\)
Answer: 1n² + 7n
Question 28
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 14\)
• The common difference: \(d = 4\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 14\)
• The common difference: \(d = 4\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 14 + (n-1) \cdot 4)}{2}\)
\(S_n = \frac{28n + 4n^2 - 4n}{2}\)
\(S_n = 2n² + 12n\)
\(S_n = \frac{28n + 4n^2 - 4n}{2}\)
\(S_n = 2n² + 12n\)
Answer: 2n² + 12n
Question 29
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 20\)
• The common difference: \(d = 4\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 20\)
• The common difference: \(d = 4\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 20 + (n-1) \cdot 4)}{2}\)
\(S_n = \frac{40n + 4n^2 - 4n}{2}\)
\(S_n = 2n² + 18n\)
\(S_n = \frac{40n + 4n^2 - 4n}{2}\)
\(S_n = 2n² + 18n\)
Answer: 2n² + 18n
Question 30
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 3\)
• The common difference: \(d = 12\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 3\)
• The common difference: \(d = 12\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 3 + (n-1) \cdot 12)}{2}\)
\(S_n = \frac{6n + 12n^2 - 12n}{2}\)
\(S_n = 6n² - 3n\)
\(S_n = \frac{6n + 12n^2 - 12n}{2}\)
\(S_n = 6n² - 3n\)
Answer: 6n² - 3n
Question 31
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 10\)
• The common difference: \(d = 14\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 10\)
• The common difference: \(d = 14\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 10 + (n-1) \cdot 14)}{2}\)
\(S_n = \frac{20n + 14n^2 - 14n}{2}\)
\(S_n = 7n² + 3n\)
\(S_n = \frac{20n + 14n^2 - 14n}{2}\)
\(S_n = 7n² + 3n\)
Answer: 7n² + 3n
Question 32
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 7\)
• The common difference: \(d = 14\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 7\)
• The common difference: \(d = 14\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 7 + (n-1) \cdot 14)}{2}\)
\(S_n = \frac{14n + 14n^2 - 14n}{2}\)
\(S_n = 7n²\)
\(S_n = \frac{14n + 14n^2 - 14n}{2}\)
\(S_n = 7n²\)
Answer: 7n²
Question 33
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 11\)
• The common difference: \(d = 6\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 11\)
• The common difference: \(d = 6\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 11 + (n-1) \cdot 6)}{2}\)
\(S_n = \frac{22n + 6n^2 - 6n}{2}\)
\(S_n = 3n² + 8n\)
\(S_n = \frac{22n + 6n^2 - 6n}{2}\)
\(S_n = 3n² + 8n\)
Answer: 3n² + 8n
Question 34
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 15\)
• The common difference: \(d = 14\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 15\)
• The common difference: \(d = 14\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 15 + (n-1) \cdot 14)}{2}\)
\(S_n = \frac{30n + 14n^2 - 14n}{2}\)
\(S_n = 7n² + 8n\)
\(S_n = \frac{30n + 14n^2 - 14n}{2}\)
\(S_n = 7n² + 8n\)
Answer: 7n² + 8n
Question 35
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 9\)
• The common difference: \(d = 10\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 9\)
• The common difference: \(d = 10\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 9 + (n-1) \cdot 10)}{2}\)
\(S_n = \frac{18n + 10n^2 - 10n}{2}\)
\(S_n = 5n² + 4n\)
\(S_n = \frac{18n + 10n^2 - 10n}{2}\)
\(S_n = 5n² + 4n\)
Answer: 5n² + 4n
Question 36
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 18\)
• The common difference: \(d = 12\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 18\)
• The common difference: \(d = 12\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 18 + (n-1) \cdot 12)}{2}\)
\(S_n = \frac{36n + 12n^2 - 12n}{2}\)
\(S_n = 6n² + 12n\)
\(S_n = \frac{36n + 12n^2 - 12n}{2}\)
\(S_n = 6n² + 12n\)
Answer: 6n² + 12n
Question 37
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 8\)
• The common difference: \(d = 12\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 8\)
• The common difference: \(d = 12\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 8 + (n-1) \cdot 12)}{2}\)
\(S_n = \frac{16n + 12n^2 - 12n}{2}\)
\(S_n = 6n² + 2n\)
\(S_n = \frac{16n + 12n^2 - 12n}{2}\)
\(S_n = 6n² + 2n\)
Answer: 6n² + 2n
Question 38
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 11\)
• The common difference: \(d = 20\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 11\)
• The common difference: \(d = 20\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 11 + (n-1) \cdot 20)}{2}\)
\(S_n = \frac{22n + 20n^2 - 20n}{2}\)
\(S_n = 10n² + 1n\)
\(S_n = \frac{22n + 20n^2 - 20n}{2}\)
\(S_n = 10n² + 1n\)
Answer: 10n² + 1n
Question 39
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 18\)
• The common difference: \(d = 2\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 18\)
• The common difference: \(d = 2\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 18 + (n-1) \cdot 2)}{2}\)
\(S_n = \frac{36n + 2n^2 - 2n}{2}\)
\(S_n = 1n² + 17n\)
\(S_n = \frac{36n + 2n^2 - 2n}{2}\)
\(S_n = 1n² + 17n\)
Answer: 1n² + 17n
Question 40
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence where:
• First term: \(a_1 = 13\)
• The common difference: \(d = 20\)
Find the sum formula \(S_n\) as a function of n.
Given an arithmetic sequence where:
• First term: \(a_1 = 13\)
• The common difference: \(d = 20\)
Find the sum formula \(S_n\) as a function of n.
Explanation:
Solution – Arithmetic Sequence:
📝 Important formulas:
\(a_n = a_1 + (n-1) \cdot d\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(a_1 + a_n)}{2} = \frac{n(2a_1 + (n-1)d)}{2}\)
🔢 Solution:\(S_n = \frac{n(2a_1 + (n-1)d)}{2}\)
\(S_n = \frac{n(2 \cdot 13 + (n-1) \cdot 20)}{2}\)
\(S_n = \frac{26n + 20n^2 - 20n}{2}\)
\(S_n = 10n² + 3n\)
\(S_n = \frac{26n + 20n^2 - 20n}{2}\)
\(S_n = 10n² + 3n\)
Answer: 10n² + 3n