Geometric Sequence — Sum of the Last k Terms — Dynamic Practice
Geometric Sequence — Sum of the Last k Terms — Dynamic Practice. Practice questions to deepen understanding of finding the sum of the last k terms in a geometric sequence. Online math practice with full solutions and step-by-step explanations.
Dynamic practice in summing the last k terms of a geometric sequence — using the total sum minus the initial partial sum, or by writing the last k terms as a new sub-sequence. New questions every attempt.
Question 1
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 7 terms, where:
• First term: \(a_1 = 16\)
• The common ratio: \(q = 2\)
Find the sum of the last 2 terms.
Given a geometric sequence with 7 terms, where:
• First term: \(a_1 = 16\)
• The common ratio: \(q = 2\)
Find the sum of the last 2 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{7} - S_{5} = 2032 - 496 = 1536\)
Answer: \(1536\)
Question 2
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 8 terms, where:
• First term: \(a_1 = 13\)
• The common ratio: \(q = 2\)
Find the sum of the last 2 terms.
Given a geometric sequence with 8 terms, where:
• First term: \(a_1 = 13\)
• The common ratio: \(q = 2\)
Find the sum of the last 2 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{8} - S_{6} = 3315 - 819 = 2496\)
Answer: \(2496\)
Question 3
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 8 terms, where:
• First term: \(a_1 = 6\)
• The common ratio: \(q = 2\)
Find the sum of the last 2 terms.
Given a geometric sequence with 8 terms, where:
• First term: \(a_1 = 6\)
• The common ratio: \(q = 2\)
Find the sum of the last 2 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{8} - S_{6} = 1530 - 378 = 1152\)
Answer: \(1152\)
Question 4
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 9 terms, where:
• First term: \(a_1 = 13\)
• The common ratio: \(q = 2\)
Find the sum of the last 2 terms.
Given a geometric sequence with 9 terms, where:
• First term: \(a_1 = 13\)
• The common ratio: \(q = 2\)
Find the sum of the last 2 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{9} - S_{7} = 6643 - 1651 = 4992\)
Answer: \(4992\)
Question 5
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 7 terms, where:
• First term: \(a_1 = 10\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
Given a geometric sequence with 7 terms, where:
• First term: \(a_1 = 10\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{7} - S_{4} = 1270 - 150 = 1120\)
Answer: \(1120\)
Question 6
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 6 terms, where:
• First term: \(a_1 = 6\)
• The common ratio: \(q = 2\)
Find the sum of the last 2 terms.
Given a geometric sequence with 6 terms, where:
• First term: \(a_1 = 6\)
• The common ratio: \(q = 2\)
Find the sum of the last 2 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{6} - S_{4} = 378 - 90 = 288\)
Answer: \(288\)
Question 7
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 7 terms, where:
• First term: \(a_1 = 6\)
• The common ratio: \(q = 2\)
Find the sum of the last 2 terms.
Given a geometric sequence with 7 terms, where:
• First term: \(a_1 = 6\)
• The common ratio: \(q = 2\)
Find the sum of the last 2 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{7} - S_{5} = 762 - 186 = 576\)
Answer: \(576\)
Question 8
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 8 terms, where:
• First term: \(a_1 = 13\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
Given a geometric sequence with 8 terms, where:
• First term: \(a_1 = 13\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{8} - S_{5} = 3315 - 403 = 2912\)
Answer: \(2912\)
Question 9
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 6 terms, where:
• First term: \(a_1 = 8\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
Given a geometric sequence with 6 terms, where:
• First term: \(a_1 = 8\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{6} - S_{3} = 504 - 56 = 448\)
Answer: \(448\)
Question 10
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 9 terms, where:
• First term: \(a_1 = 11\)
• The common ratio: \(q = 2\)
Find the sum of the last 2 terms.
Given a geometric sequence with 9 terms, where:
• First term: \(a_1 = 11\)
• The common ratio: \(q = 2\)
Find the sum of the last 2 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{9} - S_{7} = 5621 - 1397 = 4224\)
Answer: \(4224\)
Question 11
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 6 terms, where:
• First term: \(a_1 = 17\)
• The common ratio: \(q = 2\)
Find the sum of the last 2 terms.
Given a geometric sequence with 6 terms, where:
• First term: \(a_1 = 17\)
• The common ratio: \(q = 2\)
Find the sum of the last 2 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{6} - S_{4} = 1071 - 255 = 816\)
Answer: \(816\)
Question 12
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 6 terms, where:
• First term: \(a_1 = 5\)
• The common ratio: \(q = 2\)
Find the sum of the last 2 terms.
Given a geometric sequence with 6 terms, where:
• First term: \(a_1 = 5\)
• The common ratio: \(q = 2\)
Find the sum of the last 2 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{6} - S_{4} = 315 - 75 = 240\)
Answer: \(240\)
Question 13
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 6 terms, where:
• First term: \(a_1 = 12\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
Given a geometric sequence with 6 terms, where:
• First term: \(a_1 = 12\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{6} - S_{3} = 756 - 84 = 672\)
Answer: \(672\)
Question 14
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 7 terms, where:
• First term: \(a_1 = 3\)
• The common ratio: \(q = 2\)
Find the sum of the last 2 terms.
Given a geometric sequence with 7 terms, where:
• First term: \(a_1 = 3\)
• The common ratio: \(q = 2\)
Find the sum of the last 2 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{7} - S_{5} = 381 - 93 = 288\)
Answer: \(288\)
Question 15
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 6 terms, where:
• First term: \(a_1 = 15\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
Given a geometric sequence with 6 terms, where:
• First term: \(a_1 = 15\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{6} - S_{3} = 945 - 105 = 840\)
Answer: \(840\)
Question 16
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 7 terms, where:
• First term: \(a_1 = 14\)
• The common ratio: \(q = 2\)
Find the sum of the last 2 terms.
Given a geometric sequence with 7 terms, where:
• First term: \(a_1 = 14\)
• The common ratio: \(q = 2\)
Find the sum of the last 2 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{7} - S_{5} = 1778 - 434 = 1344\)
Answer: \(1344\)
Question 17
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 7 terms, where:
• First term: \(a_1 = 7\)
• The common ratio: \(q = 2\)
Find the sum of the last 2 terms.
Given a geometric sequence with 7 terms, where:
• First term: \(a_1 = 7\)
• The common ratio: \(q = 2\)
Find the sum of the last 2 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{7} - S_{5} = 889 - 217 = 672\)
Answer: \(672\)
Question 18
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 6 terms, where:
• First term: \(a_1 = 3\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
Given a geometric sequence with 6 terms, where:
• First term: \(a_1 = 3\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{6} - S_{3} = 189 - 21 = 168\)
Answer: \(168\)
Question 19
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 9 terms, where:
• First term: \(a_1 = 12\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
Given a geometric sequence with 9 terms, where:
• First term: \(a_1 = 12\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{9} - S_{6} = 6132 - 756 = 5376\)
Answer: \(5376\)
Question 20
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 9 terms, where:
• First term: \(a_1 = 11\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
Given a geometric sequence with 9 terms, where:
• First term: \(a_1 = 11\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{9} - S_{6} = 5621 - 693 = 4928\)
Answer: \(4928\)
Question 21
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 6 terms, where:
• First term: \(a_1 = 7\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
Given a geometric sequence with 6 terms, where:
• First term: \(a_1 = 7\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{6} - S_{3} = 441 - 49 = 392\)
Answer: \(392\)
Question 22
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 7 terms, where:
• First term: \(a_1 = 7\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
Given a geometric sequence with 7 terms, where:
• First term: \(a_1 = 7\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{7} - S_{4} = 889 - 105 = 784\)
Answer: \(784\)
Question 23
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 6 terms, where:
• First term: \(a_1 = 16\)
• The common ratio: \(q = 2\)
Find the sum of the last 2 terms.
Given a geometric sequence with 6 terms, where:
• First term: \(a_1 = 16\)
• The common ratio: \(q = 2\)
Find the sum of the last 2 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{6} - S_{4} = 1008 - 240 = 768\)
Answer: \(768\)
Question 24
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 7 terms, where:
• First term: \(a_1 = 4\)
• The common ratio: \(q = 2\)
Find the sum of the last 2 terms.
Given a geometric sequence with 7 terms, where:
• First term: \(a_1 = 4\)
• The common ratio: \(q = 2\)
Find the sum of the last 2 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{7} - S_{5} = 508 - 124 = 384\)
Answer: \(384\)
Question 25
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 8 terms, where:
• First term: \(a_1 = 15\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
Given a geometric sequence with 8 terms, where:
• First term: \(a_1 = 15\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{8} - S_{5} = 3825 - 465 = 3360\)
Answer: \(3360\)
Question 26
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 8 terms, where:
• First term: \(a_1 = 11\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
Given a geometric sequence with 8 terms, where:
• First term: \(a_1 = 11\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{8} - S_{5} = 2805 - 341 = 2464\)
Answer: \(2464\)
Question 27
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 8 terms, where:
• First term: \(a_1 = 12\)
• The common ratio: \(q = 2\)
Find the sum of the last 2 terms.
Given a geometric sequence with 8 terms, where:
• First term: \(a_1 = 12\)
• The common ratio: \(q = 2\)
Find the sum of the last 2 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{8} - S_{6} = 3060 - 756 = 2304\)
Answer: \(2304\)
Question 28
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 9 terms, where:
• First term: \(a_1 = 3\)
• The common ratio: \(q = 2\)
Find the sum of the last 2 terms.
Given a geometric sequence with 9 terms, where:
• First term: \(a_1 = 3\)
• The common ratio: \(q = 2\)
Find the sum of the last 2 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{9} - S_{7} = 1533 - 381 = 1152\)
Answer: \(1152\)
Question 29
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 7 terms, where:
• First term: \(a_1 = 17\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
Given a geometric sequence with 7 terms, where:
• First term: \(a_1 = 17\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{7} - S_{4} = 2159 - 255 = 1904\)
Answer: \(1904\)
Question 30
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 9 terms, where:
• First term: \(a_1 = 15\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
Given a geometric sequence with 9 terms, where:
• First term: \(a_1 = 15\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{9} - S_{6} = 7665 - 945 = 6720\)
Answer: \(6720\)
Question 31
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 7 terms, where:
• First term: \(a_1 = 12\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
Given a geometric sequence with 7 terms, where:
• First term: \(a_1 = 12\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{7} - S_{4} = 1524 - 180 = 1344\)
Answer: \(1344\)
Question 32
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 9 terms, where:
• First term: \(a_1 = 10\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
Given a geometric sequence with 9 terms, where:
• First term: \(a_1 = 10\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{9} - S_{6} = 5110 - 630 = 4480\)
Answer: \(4480\)
Question 33
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 6 terms, where:
• First term: \(a_1 = 4\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
Given a geometric sequence with 6 terms, where:
• First term: \(a_1 = 4\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{6} - S_{3} = 252 - 28 = 224\)
Answer: \(224\)
Question 34
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 7 terms, where:
• First term: \(a_1 = 16\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
Given a geometric sequence with 7 terms, where:
• First term: \(a_1 = 16\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{7} - S_{4} = 2032 - 240 = 1792\)
Answer: \(1792\)
Question 35
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 8 terms, where:
• First term: \(a_1 = 2\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
Given a geometric sequence with 8 terms, where:
• First term: \(a_1 = 2\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{8} - S_{5} = 510 - 62 = 448\)
Answer: \(448\)
Question 36
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 7 terms, where:
• First term: \(a_1 = 6\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
Given a geometric sequence with 7 terms, where:
• First term: \(a_1 = 6\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{7} - S_{4} = 762 - 90 = 672\)
Answer: \(672\)
Question 37
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 7 terms, where:
• First term: \(a_1 = 15\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
Given a geometric sequence with 7 terms, where:
• First term: \(a_1 = 15\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{7} - S_{4} = 1905 - 225 = 1680\)
Answer: \(1680\)
Question 38
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 7 terms, where:
• First term: \(a_1 = 3\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
Given a geometric sequence with 7 terms, where:
• First term: \(a_1 = 3\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{7} - S_{4} = 381 - 45 = 336\)
Answer: \(336\)
Question 39
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 7 terms, where:
• First term: \(a_1 = 8\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
Given a geometric sequence with 7 terms, where:
• First term: \(a_1 = 8\)
• The common ratio: \(q = 2\)
Find the sum of the last 3 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{7} - S_{4} = 1016 - 120 = 896\)
Answer: \(896\)
Question 40
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence with 6 terms, where:
• First term: \(a_1 = 14\)
• The common ratio: \(q = 2\)
Find the sum of the last 2 terms.
Given a geometric sequence with 6 terms, where:
• First term: \(a_1 = 14\)
• The common ratio: \(q = 2\)
Find the sum of the last 2 terms.
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:
Sum of the last k terms = \(S_n - S_{n-k}\)
Sum of the last k terms = \(S_n - S_{n-k}\)
\(S_{6} - S_{4} = 882 - 210 = 672\)
Answer: \(672\)