Arithmetic Sequence — Sum of Last k Terms — Dynamic Practice (Part 2)
Arithmetic Sequence — Sum of Last k Terms — Dynamic Practice (Part 2). Practice questions to deepen understanding of finding the sum of the last k terms in an arithmetic sequence — advanced variations. Online math practice with full solutions and step-by-step explanations.
Dynamic advanced practice in summing the last k terms — by reversing the sequence or by total sum minus initial partial sum. New questions every attempt.
Question 1
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 23 terms, where:
• First term: \(a_1 = 5\)
• The common difference: \(d = 1\)
Find the sum of the last 4 terms.
Given an arithmetic sequence with 23 terms, where:
• First term: \(a_1 = 5\)
• The common difference: \(d = 1\)
Find the sum of the last 4 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 4 last terms = \(S_{23} - S_{19}\) = 102
Answer: 102
Question 2
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 19 terms, where:
• First term: \(a_1 = 7\)
• The common difference: \(d = 1\)
Find the sum of the last 5 terms.
Given an arithmetic sequence with 19 terms, where:
• First term: \(a_1 = 7\)
• The common difference: \(d = 1\)
Find the sum of the last 5 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 5 last terms = \(S_{19} - S_{14}\) = 115
Answer: 115
Question 3
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 21 terms, where:
• First term: \(a_1 = 3\)
• The common difference: \(d = 1\)
Find the sum of the last 4 terms.
Given an arithmetic sequence with 21 terms, where:
• First term: \(a_1 = 3\)
• The common difference: \(d = 1\)
Find the sum of the last 4 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 4 last terms = \(S_{21} - S_{17}\) = 86
Answer: 86
Question 4
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 16 terms, where:
• First term: \(a_1 = 2\)
• The common difference: \(d = 4\)
Find the sum of the last 6 terms.
Given an arithmetic sequence with 16 terms, where:
• First term: \(a_1 = 2\)
• The common difference: \(d = 4\)
Find the sum of the last 6 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 6 last terms = \(S_{16} - S_{10}\) = 312
Answer: 312
Question 5
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 21 terms, where:
• First term: \(a_1 = 7\)
• The common difference: \(d = 2\)
Find the sum of the last 7 terms.
Given an arithmetic sequence with 21 terms, where:
• First term: \(a_1 = 7\)
• The common difference: \(d = 2\)
Find the sum of the last 7 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 7 last terms = \(S_{21} - S_{14}\) = 287
Answer: 287
Question 6
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 16 terms, where:
• First term: \(a_1 = 3\)
• The common difference: \(d = 1\)
Find the sum of the last 3 terms.
Given an arithmetic sequence with 16 terms, where:
• First term: \(a_1 = 3\)
• The common difference: \(d = 1\)
Find the sum of the last 3 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 3 last terms = \(S_{16} - S_{13}\) = 51
Answer: 51
Question 7
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 15 terms, where:
• First term: \(a_1 = 2\)
• The common difference: \(d = 3\)
Find the sum of the last 3 terms.
Given an arithmetic sequence with 15 terms, where:
• First term: \(a_1 = 2\)
• The common difference: \(d = 3\)
Find the sum of the last 3 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 3 last terms = \(S_{15} - S_{12}\) = 123
Answer: 123
Question 8
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 17 terms, where:
• First term: \(a_1 = 6\)
• The common difference: \(d = 3\)
Find the sum of the last 5 terms.
Given an arithmetic sequence with 17 terms, where:
• First term: \(a_1 = 6\)
• The common difference: \(d = 3\)
Find the sum of the last 5 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 5 last terms = \(S_{17} - S_{12}\) = 240
Answer: 240
Question 9
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 15 terms, where:
• First term: \(a_1 = 2\)
• The common difference: \(d = 4\)
Find the sum of the last 7 terms.
Given an arithmetic sequence with 15 terms, where:
• First term: \(a_1 = 2\)
• The common difference: \(d = 4\)
Find the sum of the last 7 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 7 last terms = \(S_{15} - S_{8}\) = 322
Answer: 322
Question 10
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 24 terms, where:
• First term: \(a_1 = 7\)
• The common difference: \(d = 1\)
Find the sum of the last 5 terms.
Given an arithmetic sequence with 24 terms, where:
• First term: \(a_1 = 7\)
• The common difference: \(d = 1\)
Find the sum of the last 5 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 5 last terms = \(S_{24} - S_{19}\) = 140
Answer: 140
Question 11
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 23 terms, where:
• First term: \(a_1 = 3\)
• The common difference: \(d = 2\)
Find the sum of the last 7 terms.
Given an arithmetic sequence with 23 terms, where:
• First term: \(a_1 = 3\)
• The common difference: \(d = 2\)
Find the sum of the last 7 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 7 last terms = \(S_{23} - S_{16}\) = 287
Answer: 287
Question 12
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 20 terms, where:
• First term: \(a_1 = 8\)
• The common difference: \(d = 1\)
Find the sum of the last 4 terms.
Given an arithmetic sequence with 20 terms, where:
• First term: \(a_1 = 8\)
• The common difference: \(d = 1\)
Find the sum of the last 4 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 4 last terms = \(S_{20} - S_{16}\) = 102
Answer: 102
Question 13
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 18 terms, where:
• First term: \(a_1 = 4\)
• The common difference: \(d = 2\)
Find the sum of the last 6 terms.
Given an arithmetic sequence with 18 terms, where:
• First term: \(a_1 = 4\)
• The common difference: \(d = 2\)
Find the sum of the last 6 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 6 last terms = \(S_{18} - S_{12}\) = 198
Answer: 198
Question 14
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 19 terms, where:
• First term: \(a_1 = 9\)
• The common difference: \(d = 4\)
Find the sum of the last 6 terms.
Given an arithmetic sequence with 19 terms, where:
• First term: \(a_1 = 9\)
• The common difference: \(d = 4\)
Find the sum of the last 6 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 6 last terms = \(S_{19} - S_{13}\) = 426
Answer: 426
Question 15
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 15 terms, where:
• First term: \(a_1 = 5\)
• The common difference: \(d = 1\)
Find the sum of the last 6 terms.
Given an arithmetic sequence with 15 terms, where:
• First term: \(a_1 = 5\)
• The common difference: \(d = 1\)
Find the sum of the last 6 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 6 last terms = \(S_{15} - S_{9}\) = 99
Answer: 99
Question 16
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 15 terms, where:
• First term: \(a_1 = 8\)
• The common difference: \(d = 2\)
Find the sum of the last 5 terms.
Given an arithmetic sequence with 15 terms, where:
• First term: \(a_1 = 8\)
• The common difference: \(d = 2\)
Find the sum of the last 5 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 5 last terms = \(S_{15} - S_{10}\) = 160
Answer: 160
Question 17
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 20 terms, where:
• First term: \(a_1 = 4\)
• The common difference: \(d = 3\)
Find the sum of the last 5 terms.
Given an arithmetic sequence with 20 terms, where:
• First term: \(a_1 = 4\)
• The common difference: \(d = 3\)
Find the sum of the last 5 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 5 last terms = \(S_{20} - S_{15}\) = 275
Answer: 275
Question 18
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 19 terms, where:
• First term: \(a_1 = 9\)
• The common difference: \(d = 4\)
Find the sum of the last 5 terms.
Given an arithmetic sequence with 19 terms, where:
• First term: \(a_1 = 9\)
• The common difference: \(d = 4\)
Find the sum of the last 5 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 5 last terms = \(S_{19} - S_{14}\) = 365
Answer: 365
Question 19
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 24 terms, where:
• First term: \(a_1 = 7\)
• The common difference: \(d = 2\)
Find the sum of the last 5 terms.
Given an arithmetic sequence with 24 terms, where:
• First term: \(a_1 = 7\)
• The common difference: \(d = 2\)
Find the sum of the last 5 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 5 last terms = \(S_{24} - S_{19}\) = 245
Answer: 245
Question 20
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 23 terms, where:
• First term: \(a_1 = 6\)
• The common difference: \(d = 1\)
Find the sum of the last 5 terms.
Given an arithmetic sequence with 23 terms, where:
• First term: \(a_1 = 6\)
• The common difference: \(d = 1\)
Find the sum of the last 5 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 5 last terms = \(S_{23} - S_{18}\) = 130
Answer: 130
Question 21
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 20 terms, where:
• First term: \(a_1 = 4\)
• The common difference: \(d = 4\)
Find the sum of the last 7 terms.
Given an arithmetic sequence with 20 terms, where:
• First term: \(a_1 = 4\)
• The common difference: \(d = 4\)
Find the sum of the last 7 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 7 last terms = \(S_{20} - S_{13}\) = 476
Answer: 476
Question 22
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 17 terms, where:
• First term: \(a_1 = 7\)
• The common difference: \(d = 3\)
Find the sum of the last 6 terms.
Given an arithmetic sequence with 17 terms, where:
• First term: \(a_1 = 7\)
• The common difference: \(d = 3\)
Find the sum of the last 6 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 6 last terms = \(S_{17} - S_{11}\) = 285
Answer: 285
Question 23
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 16 terms, where:
• First term: \(a_1 = 5\)
• The common difference: \(d = 1\)
Find the sum of the last 4 terms.
Given an arithmetic sequence with 16 terms, where:
• First term: \(a_1 = 5\)
• The common difference: \(d = 1\)
Find the sum of the last 4 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 4 last terms = \(S_{16} - S_{12}\) = 74
Answer: 74
Question 24
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 19 terms, where:
• First term: \(a_1 = 7\)
• The common difference: \(d = 3\)
Find the sum of the last 4 terms.
Given an arithmetic sequence with 19 terms, where:
• First term: \(a_1 = 7\)
• The common difference: \(d = 3\)
Find the sum of the last 4 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 4 last terms = \(S_{19} - S_{15}\) = 226
Answer: 226
Question 25
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 19 terms, where:
• First term: \(a_1 = 8\)
• The common difference: \(d = 3\)
Find the sum of the last 6 terms.
Given an arithmetic sequence with 19 terms, where:
• First term: \(a_1 = 8\)
• The common difference: \(d = 3\)
Find the sum of the last 6 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 6 last terms = \(S_{19} - S_{13}\) = 327
Answer: 327
Question 26
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 19 terms, where:
• First term: \(a_1 = 8\)
• The common difference: \(d = 2\)
Find the sum of the last 7 terms.
Given an arithmetic sequence with 19 terms, where:
• First term: \(a_1 = 8\)
• The common difference: \(d = 2\)
Find the sum of the last 7 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 7 last terms = \(S_{19} - S_{12}\) = 266
Answer: 266
Question 27
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 18 terms, where:
• First term: \(a_1 = 7\)
• The common difference: \(d = 3\)
Find the sum of the last 3 terms.
Given an arithmetic sequence with 18 terms, where:
• First term: \(a_1 = 7\)
• The common difference: \(d = 3\)
Find the sum of the last 3 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 3 last terms = \(S_{18} - S_{15}\) = 165
Answer: 165
Question 28
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 15 terms, where:
• First term: \(a_1 = 3\)
• The common difference: \(d = 4\)
Find the sum of the last 3 terms.
Given an arithmetic sequence with 15 terms, where:
• First term: \(a_1 = 3\)
• The common difference: \(d = 4\)
Find the sum of the last 3 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 3 last terms = \(S_{15} - S_{12}\) = 165
Answer: 165
Question 29
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 20 terms, where:
• First term: \(a_1 = 5\)
• The common difference: \(d = 1\)
Find the sum of the last 7 terms.
Given an arithmetic sequence with 20 terms, where:
• First term: \(a_1 = 5\)
• The common difference: \(d = 1\)
Find the sum of the last 7 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 7 last terms = \(S_{20} - S_{13}\) = 147
Answer: 147
Question 30
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 19 terms, where:
• First term: \(a_1 = 1\)
• The common difference: \(d = 4\)
Find the sum of the last 4 terms.
Given an arithmetic sequence with 19 terms, where:
• First term: \(a_1 = 1\)
• The common difference: \(d = 4\)
Find the sum of the last 4 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 4 last terms = \(S_{19} - S_{15}\) = 268
Answer: 268
Question 31
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 16 terms, where:
• First term: \(a_1 = 1\)
• The common difference: \(d = 4\)
Find the sum of the last 6 terms.
Given an arithmetic sequence with 16 terms, where:
• First term: \(a_1 = 1\)
• The common difference: \(d = 4\)
Find the sum of the last 6 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 6 last terms = \(S_{16} - S_{10}\) = 306
Answer: 306
Question 32
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 18 terms, where:
• First term: \(a_1 = 1\)
• The common difference: \(d = 3\)
Find the sum of the last 4 terms.
Given an arithmetic sequence with 18 terms, where:
• First term: \(a_1 = 1\)
• The common difference: \(d = 3\)
Find the sum of the last 4 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 4 last terms = \(S_{18} - S_{14}\) = 190
Answer: 190
Question 33
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 22 terms, where:
• First term: \(a_1 = 1\)
• The common difference: \(d = 2\)
Find the sum of the last 7 terms.
Given an arithmetic sequence with 22 terms, where:
• First term: \(a_1 = 1\)
• The common difference: \(d = 2\)
Find the sum of the last 7 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 7 last terms = \(S_{22} - S_{15}\) = 259
Answer: 259
Question 34
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 23 terms, where:
• First term: \(a_1 = 8\)
• The common difference: \(d = 3\)
Find the sum of the last 5 terms.
Given an arithmetic sequence with 23 terms, where:
• First term: \(a_1 = 8\)
• The common difference: \(d = 3\)
Find the sum of the last 5 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 5 last terms = \(S_{23} - S_{18}\) = 340
Answer: 340
Question 35
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 20 terms, where:
• First term: \(a_1 = 6\)
• The common difference: \(d = 4\)
Find the sum of the last 4 terms.
Given an arithmetic sequence with 20 terms, where:
• First term: \(a_1 = 6\)
• The common difference: \(d = 4\)
Find the sum of the last 4 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 4 last terms = \(S_{20} - S_{16}\) = 304
Answer: 304
Question 36
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 17 terms, where:
• First term: \(a_1 = 9\)
• The common difference: \(d = 2\)
Find the sum of the last 7 terms.
Given an arithmetic sequence with 17 terms, where:
• First term: \(a_1 = 9\)
• The common difference: \(d = 2\)
Find the sum of the last 7 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 7 last terms = \(S_{17} - S_{10}\) = 245
Answer: 245
Question 37
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 18 terms, where:
• First term: \(a_1 = 8\)
• The common difference: \(d = 2\)
Find the sum of the last 7 terms.
Given an arithmetic sequence with 18 terms, where:
• First term: \(a_1 = 8\)
• The common difference: \(d = 2\)
Find the sum of the last 7 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 7 last terms = \(S_{18} - S_{11}\) = 252
Answer: 252
Question 38
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 20 terms, where:
• First term: \(a_1 = 5\)
• The common difference: \(d = 3\)
Find the sum of the last 5 terms.
Given an arithmetic sequence with 20 terms, where:
• First term: \(a_1 = 5\)
• The common difference: \(d = 3\)
Find the sum of the last 5 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 5 last terms = \(S_{20} - S_{15}\) = 280
Answer: 280
Question 39
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 19 terms, where:
• First term: \(a_1 = 6\)
• The common difference: \(d = 1\)
Find the sum of the last 4 terms.
Given an arithmetic sequence with 19 terms, where:
• First term: \(a_1 = 6\)
• The common difference: \(d = 1\)
Find the sum of the last 4 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 4 last terms = \(S_{19} - S_{15}\) = 90
Answer: 90
Question 40
2.50 pts
📊 Arithmetic Sequence:
Given an arithmetic sequence with 22 terms, where:
• First term: \(a_1 = 3\)
• The common difference: \(d = 3\)
Find the sum of the last 4 terms.
Given an arithmetic sequence with 22 terms, where:
• First term: \(a_1 = 3\)
• The common difference: \(d = 3\)
Find the sum of the last 4 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of 4 last terms = \(S_{22} - S_{18}\) = 246
Answer: 246