Geometric Sequence — Sum Formula Sₙ — Dynamic Practice (Part 2)
Geometric Sequence — Sum Formula Sₙ — Dynamic Practice (Part 2). Practice questions to deepen understanding of deriving the sum formula Sₙ in a geometric sequence — advanced variations. Online math practice with full solutions and step-by-step explanations.
Dynamic advanced practice in deriving Sₙ — solving systems for a₁ and q to obtain Sₙ = a₁ · (qⁿ − 1)/(q − 1). New questions every attempt.
Question 1
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 23\)
• The common ratio: \(q = 4\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 23\)
• The common ratio: \(q = 4\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{23(4^n - 1)}{4 - 1} = \frac{23(4^n - 1)}{3}\)
Answer: \(\frac{23(4^n - 1)}{3}\)
Question 2
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 9\)
• The common ratio: \(q = 3\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 9\)
• The common ratio: \(q = 3\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{9(3^n - 1)}{3 - 1} = \frac{9(3^n - 1)}{2}\)
Answer: \(\frac{9(3^n - 1)}{2}\)
Question 3
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 23\)
• The common ratio: \(q = 2\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 23\)
• The common ratio: \(q = 2\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{23(2^n - 1)}{2 - 1} = \frac{23(2^n - 1)}{1}\)
Answer: \(\frac{23(2^n - 1)}{1}\)
Question 4
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 12\)
• The common ratio: \(q = 2\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 12\)
• The common ratio: \(q = 2\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{12(2^n - 1)}{2 - 1} = \frac{12(2^n - 1)}{1}\)
Answer: \(\frac{12(2^n - 1)}{1}\)
Question 5
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 21\)
• The common ratio: \(q = 4\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 21\)
• The common ratio: \(q = 4\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{21(4^n - 1)}{4 - 1} = \frac{21(4^n - 1)}{3}\)
Answer: \(\frac{21(4^n - 1)}{3}\)
Question 6
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 26\)
• The common ratio: \(q = 5\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 26\)
• The common ratio: \(q = 5\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{26(5^n - 1)}{5 - 1} = \frac{26(5^n - 1)}{4}\)
Answer: \(\frac{26(5^n - 1)}{4}\)
Question 7
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 17\)
• The common ratio: \(q = 3\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 17\)
• The common ratio: \(q = 3\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{17(3^n - 1)}{3 - 1} = \frac{17(3^n - 1)}{2}\)
Answer: \(\frac{17(3^n - 1)}{2}\)
Question 8
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 22\)
• The common ratio: \(q = 4\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 22\)
• The common ratio: \(q = 4\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{22(4^n - 1)}{4 - 1} = \frac{22(4^n - 1)}{3}\)
Answer: \(\frac{22(4^n - 1)}{3}\)
Question 9
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 9\)
• The common ratio: \(q = 4\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 9\)
• The common ratio: \(q = 4\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{9(4^n - 1)}{4 - 1} = \frac{9(4^n - 1)}{3}\)
Answer: \(\frac{9(4^n - 1)}{3}\)
Question 10
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 13\)
• The common ratio: \(q = 4\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 13\)
• The common ratio: \(q = 4\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{13(4^n - 1)}{4 - 1} = \frac{13(4^n - 1)}{3}\)
Answer: \(\frac{13(4^n - 1)}{3}\)
Question 11
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 19\)
• The common ratio: \(q = 3\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 19\)
• The common ratio: \(q = 3\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{19(3^n - 1)}{3 - 1} = \frac{19(3^n - 1)}{2}\)
Answer: \(\frac{19(3^n - 1)}{2}\)
Question 12
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 29\)
• The common ratio: \(q = 3\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 29\)
• The common ratio: \(q = 3\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{29(3^n - 1)}{3 - 1} = \frac{29(3^n - 1)}{2}\)
Answer: \(\frac{29(3^n - 1)}{2}\)
Question 13
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 14\)
• The common ratio: \(q = 3\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 14\)
• The common ratio: \(q = 3\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{14(3^n - 1)}{3 - 1} = \frac{14(3^n - 1)}{2}\)
Answer: \(\frac{14(3^n - 1)}{2}\)
Question 14
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 3\)
• The common ratio: \(q = 2\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 3\)
• The common ratio: \(q = 2\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{3(2^n - 1)}{2 - 1} = \frac{3(2^n - 1)}{1}\)
Answer: \(\frac{3(2^n - 1)}{1}\)
Question 15
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 8\)
• The common ratio: \(q = 4\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 8\)
• The common ratio: \(q = 4\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{8(4^n - 1)}{4 - 1} = \frac{8(4^n - 1)}{3}\)
Answer: \(\frac{8(4^n - 1)}{3}\)
Question 16
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 12\)
• The common ratio: \(q = 3\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 12\)
• The common ratio: \(q = 3\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{12(3^n - 1)}{3 - 1} = \frac{12(3^n - 1)}{2}\)
Answer: \(\frac{12(3^n - 1)}{2}\)
Question 17
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 15\)
• The common ratio: \(q = 4\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 15\)
• The common ratio: \(q = 4\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{15(4^n - 1)}{4 - 1} = \frac{15(4^n - 1)}{3}\)
Answer: \(\frac{15(4^n - 1)}{3}\)
Question 18
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 5\)
• The common ratio: \(q = 3\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 5\)
• The common ratio: \(q = 3\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{5(3^n - 1)}{3 - 1} = \frac{5(3^n - 1)}{2}\)
Answer: \(\frac{5(3^n - 1)}{2}\)
Question 19
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 4\)
• The common ratio: \(q = 5\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 4\)
• The common ratio: \(q = 5\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{4(5^n - 1)}{5 - 1} = \frac{4(5^n - 1)}{4}\)
Answer: \(\frac{4(5^n - 1)}{4}\)
Question 20
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 18\)
• The common ratio: \(q = 5\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 18\)
• The common ratio: \(q = 5\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{18(5^n - 1)}{5 - 1} = \frac{18(5^n - 1)}{4}\)
Answer: \(\frac{18(5^n - 1)}{4}\)
Question 21
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 29\)
• The common ratio: \(q = 2\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 29\)
• The common ratio: \(q = 2\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{29(2^n - 1)}{2 - 1} = \frac{29(2^n - 1)}{1}\)
Answer: \(\frac{29(2^n - 1)}{1}\)
Question 22
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 28\)
• The common ratio: \(q = 5\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 28\)
• The common ratio: \(q = 5\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{28(5^n - 1)}{5 - 1} = \frac{28(5^n - 1)}{4}\)
Answer: \(\frac{28(5^n - 1)}{4}\)
Question 23
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 7\)
• The common ratio: \(q = 3\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 7\)
• The common ratio: \(q = 3\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{7(3^n - 1)}{3 - 1} = \frac{7(3^n - 1)}{2}\)
Answer: \(\frac{7(3^n - 1)}{2}\)
Question 24
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 3\)
• The common ratio: \(q = 5\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 3\)
• The common ratio: \(q = 5\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{3(5^n - 1)}{5 - 1} = \frac{3(5^n - 1)}{4}\)
Answer: \(\frac{3(5^n - 1)}{4}\)
Question 25
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 30\)
• The common ratio: \(q = 4\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 30\)
• The common ratio: \(q = 4\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{30(4^n - 1)}{4 - 1} = \frac{30(4^n - 1)}{3}\)
Answer: \(\frac{30(4^n - 1)}{3}\)
Question 26
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 19\)
• The common ratio: \(q = 4\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 19\)
• The common ratio: \(q = 4\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{19(4^n - 1)}{4 - 1} = \frac{19(4^n - 1)}{3}\)
Answer: \(\frac{19(4^n - 1)}{3}\)
Question 27
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 6\)
• The common ratio: \(q = 4\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 6\)
• The common ratio: \(q = 4\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{6(4^n - 1)}{4 - 1} = \frac{6(4^n - 1)}{3}\)
Answer: \(\frac{6(4^n - 1)}{3}\)
Question 28
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 27\)
• The common ratio: \(q = 2\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 27\)
• The common ratio: \(q = 2\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{27(2^n - 1)}{2 - 1} = \frac{27(2^n - 1)}{1}\)
Answer: \(\frac{27(2^n - 1)}{1}\)
Question 29
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 17\)
• The common ratio: \(q = 5\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 17\)
• The common ratio: \(q = 5\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{17(5^n - 1)}{5 - 1} = \frac{17(5^n - 1)}{4}\)
Answer: \(\frac{17(5^n - 1)}{4}\)
Question 30
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 2\)
• The common ratio: \(q = 5\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 2\)
• The common ratio: \(q = 5\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{2(5^n - 1)}{5 - 1} = \frac{2(5^n - 1)}{4}\)
Answer: \(\frac{2(5^n - 1)}{4}\)
Question 31
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 30\)
• The common ratio: \(q = 2\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 30\)
• The common ratio: \(q = 2\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{30(2^n - 1)}{2 - 1} = \frac{30(2^n - 1)}{1}\)
Answer: \(\frac{30(2^n - 1)}{1}\)
Question 32
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 6\)
• The common ratio: \(q = 2\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 6\)
• The common ratio: \(q = 2\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{6(2^n - 1)}{2 - 1} = \frac{6(2^n - 1)}{1}\)
Answer: \(\frac{6(2^n - 1)}{1}\)
Question 33
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 4\)
• The common ratio: \(q = 2\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 4\)
• The common ratio: \(q = 2\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{4(2^n - 1)}{2 - 1} = \frac{4(2^n - 1)}{1}\)
Answer: \(\frac{4(2^n - 1)}{1}\)
Question 34
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 4\)
• The common ratio: \(q = 4\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 4\)
• The common ratio: \(q = 4\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{4(4^n - 1)}{4 - 1} = \frac{4(4^n - 1)}{3}\)
Answer: \(\frac{4(4^n - 1)}{3}\)
Question 35
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 13\)
• The common ratio: \(q = 5\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 13\)
• The common ratio: \(q = 5\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{13(5^n - 1)}{5 - 1} = \frac{13(5^n - 1)}{4}\)
Answer: \(\frac{13(5^n - 1)}{4}\)
Question 36
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 11\)
• The common ratio: \(q = 3\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 11\)
• The common ratio: \(q = 3\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{11(3^n - 1)}{3 - 1} = \frac{11(3^n - 1)}{2}\)
Answer: \(\frac{11(3^n - 1)}{2}\)
Question 37
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 29\)
• The common ratio: \(q = 4\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 29\)
• The common ratio: \(q = 4\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{29(4^n - 1)}{4 - 1} = \frac{29(4^n - 1)}{3}\)
Answer: \(\frac{29(4^n - 1)}{3}\)
Question 38
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 25\)
• The common ratio: \(q = 5\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 25\)
• The common ratio: \(q = 5\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{25(5^n - 1)}{5 - 1} = \frac{25(5^n - 1)}{4}\)
Answer: \(\frac{25(5^n - 1)}{4}\)
Question 39
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 21\)
• The common ratio: \(q = 3\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 21\)
• The common ratio: \(q = 3\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{21(3^n - 1)}{3 - 1} = \frac{21(3^n - 1)}{2}\)
Answer: \(\frac{21(3^n - 1)}{2}\)
Question 40
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence where:
• First term: \(a_1 = 15\)
• The common ratio: \(q = 3\)
Find the sum formula \(S_n\).
Given a geometric sequence where:
• First term: \(a_1 = 15\)
• The common ratio: \(q = 3\)
Find the sum formula \(S_n\).
Explanation:
Solution – Geometric Sequence:
📝 Important formulas:
\(a_n = a_1 \cdot q^{n-1}\)
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
\(S_n = \frac{a_1(q^n - 1)}{q - 1}\) (when \(q \neq 1\))
\(S_\infty = \frac{a_1}{1-q}\) (when \(|q| < 1\))
🔢 Solution:
The formula for the sum of n terms:
The formula for the sum of n terms:
\(S_n = \frac{15(3^n - 1)}{3 - 1} = \frac{15(3^n - 1)}{2}\)
Answer: \(\frac{15(3^n - 1)}{2}\)