Geometric Sequence — Building the General Term Formula — Dynamic Practice
Geometric Sequence — Building the General Term Formula — Dynamic Practice. Practice questions to deepen understanding of building the formula for the general term in a geometric sequence. Online math practice with full solutions and detailed explanations.
Dynamic practice in building the formula aₙ — solve for a₁ and q from given data, then write aₙ = a₁ · q^(n−1). New questions every attempt.
Question 1
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
11, 44, 176, 704, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 4\), \(a_1 = 11\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 11 \cdot 4^{n-1}\)
From the terms we find: \(q = 4\), \(a_1 = 11\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 11 \cdot 4^{n-1}\)
Answer: \(11 \cdot 4^{n-1}\)
Question 2
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
31, 93, 279, 837, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 3\), \(a_1 = 31\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 31 \cdot 3^{n-1}\)
From the terms we find: \(q = 3\), \(a_1 = 31\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 31 \cdot 3^{n-1}\)
Answer: \(31 \cdot 3^{n-1}\)
Question 3
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
9, 45, 225, 1125, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 5\), \(a_1 = 9\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 9 \cdot 5^{n-1}\)
From the terms we find: \(q = 5\), \(a_1 = 9\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 9 \cdot 5^{n-1}\)
Answer: \(9 \cdot 5^{n-1}\)
Question 4
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
14, 56, 224, 896, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 4\), \(a_1 = 14\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 14 \cdot 4^{n-1}\)
From the terms we find: \(q = 4\), \(a_1 = 14\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 14 \cdot 4^{n-1}\)
Answer: \(14 \cdot 4^{n-1}\)
Question 5
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
7, 21, 63, 189, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 3\), \(a_1 = 7\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 7 \cdot 3^{n-1}\)
From the terms we find: \(q = 3\), \(a_1 = 7\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 7 \cdot 3^{n-1}\)
Answer: \(7 \cdot 3^{n-1}\)
Question 6
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
26, 78, 234, 702, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 3\), \(a_1 = 26\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 26 \cdot 3^{n-1}\)
From the terms we find: \(q = 3\), \(a_1 = 26\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 26 \cdot 3^{n-1}\)
Answer: \(26 \cdot 3^{n-1}\)
Question 7
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
29, 58, 116, 232, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 2\), \(a_1 = 29\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 29 \cdot 2^{n-1}\)
From the terms we find: \(q = 2\), \(a_1 = 29\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 29 \cdot 2^{n-1}\)
Answer: \(29 \cdot 2^{n-1}\)
Question 8
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
30, 90, 270, 810, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 3\), \(a_1 = 30\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 30 \cdot 3^{n-1}\)
From the terms we find: \(q = 3\), \(a_1 = 30\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 30 \cdot 3^{n-1}\)
Answer: \(30 \cdot 3^{n-1}\)
Question 9
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
23, 46, 92, 184, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 2\), \(a_1 = 23\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 23 \cdot 2^{n-1}\)
From the terms we find: \(q = 2\), \(a_1 = 23\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 23 \cdot 2^{n-1}\)
Answer: \(23 \cdot 2^{n-1}\)
Question 10
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
8, 16, 32, 64, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 2\), \(a_1 = 8\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 8 \cdot 2^{n-1}\)
From the terms we find: \(q = 2\), \(a_1 = 8\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 8 \cdot 2^{n-1}\)
Answer: \(8 \cdot 2^{n-1}\)
Question 11
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
18, 72, 288, 1152, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 4\), \(a_1 = 18\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 18 \cdot 4^{n-1}\)
From the terms we find: \(q = 4\), \(a_1 = 18\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 18 \cdot 4^{n-1}\)
Answer: \(18 \cdot 4^{n-1}\)
Question 12
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
12, 24, 48, 96, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 2\), \(a_1 = 12\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 12 \cdot 2^{n-1}\)
From the terms we find: \(q = 2\), \(a_1 = 12\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 12 \cdot 2^{n-1}\)
Answer: \(12 \cdot 2^{n-1}\)
Question 13
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
26, 130, 650, 3250, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 5\), \(a_1 = 26\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 26 \cdot 5^{n-1}\)
From the terms we find: \(q = 5\), \(a_1 = 26\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 26 \cdot 5^{n-1}\)
Answer: \(26 \cdot 5^{n-1}\)
Question 14
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
13, 39, 117, 351, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 3\), \(a_1 = 13\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 13 \cdot 3^{n-1}\)
From the terms we find: \(q = 3\), \(a_1 = 13\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 13 \cdot 3^{n-1}\)
Answer: \(13 \cdot 3^{n-1}\)
Question 15
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
13, 26, 52, 104, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 2\), \(a_1 = 13\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 13 \cdot 2^{n-1}\)
From the terms we find: \(q = 2\), \(a_1 = 13\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 13 \cdot 2^{n-1}\)
Answer: \(13 \cdot 2^{n-1}\)
Question 16
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
14, 28, 56, 112, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 2\), \(a_1 = 14\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 14 \cdot 2^{n-1}\)
From the terms we find: \(q = 2\), \(a_1 = 14\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 14 \cdot 2^{n-1}\)
Answer: \(14 \cdot 2^{n-1}\)
Question 17
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
5, 20, 80, 320, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 4\), \(a_1 = 5\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 5 \cdot 4^{n-1}\)
From the terms we find: \(q = 4\), \(a_1 = 5\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 5 \cdot 4^{n-1}\)
Answer: \(5 \cdot 4^{n-1}\)
Question 18
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
24, 72, 216, 648, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 3\), \(a_1 = 24\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 24 \cdot 3^{n-1}\)
From the terms we find: \(q = 3\), \(a_1 = 24\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 24 \cdot 3^{n-1}\)
Answer: \(24 \cdot 3^{n-1}\)
Question 19
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
6, 18, 54, 162, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 3\), \(a_1 = 6\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 6 \cdot 3^{n-1}\)
From the terms we find: \(q = 3\), \(a_1 = 6\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 6 \cdot 3^{n-1}\)
Answer: \(6 \cdot 3^{n-1}\)
Question 20
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
16, 64, 256, 1024, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 4\), \(a_1 = 16\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 16 \cdot 4^{n-1}\)
From the terms we find: \(q = 4\), \(a_1 = 16\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 16 \cdot 4^{n-1}\)
Answer: \(16 \cdot 4^{n-1}\)
Question 21
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
28, 84, 252, 756, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 3\), \(a_1 = 28\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 28 \cdot 3^{n-1}\)
From the terms we find: \(q = 3\), \(a_1 = 28\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 28 \cdot 3^{n-1}\)
Answer: \(28 \cdot 3^{n-1}\)
Question 22
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
14, 70, 350, 1750, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 5\), \(a_1 = 14\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 14 \cdot 5^{n-1}\)
From the terms we find: \(q = 5\), \(a_1 = 14\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 14 \cdot 5^{n-1}\)
Answer: \(14 \cdot 5^{n-1}\)
Question 23
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
2, 8, 32, 128, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 4\), \(a_1 = 2\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 2 \cdot 4^{n-1}\)
From the terms we find: \(q = 4\), \(a_1 = 2\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 2 \cdot 4^{n-1}\)
Answer: \(2 \cdot 4^{n-1}\)
Question 24
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
5, 15, 45, 135, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 3\), \(a_1 = 5\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 5 \cdot 3^{n-1}\)
From the terms we find: \(q = 3\), \(a_1 = 5\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 5 \cdot 3^{n-1}\)
Answer: \(5 \cdot 3^{n-1}\)
Question 25
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
26, 104, 416, 1664, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 4\), \(a_1 = 26\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 26 \cdot 4^{n-1}\)
From the terms we find: \(q = 4\), \(a_1 = 26\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 26 \cdot 4^{n-1}\)
Answer: \(26 \cdot 4^{n-1}\)
Question 26
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
29, 87, 261, 783, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 3\), \(a_1 = 29\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 29 \cdot 3^{n-1}\)
From the terms we find: \(q = 3\), \(a_1 = 29\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 29 \cdot 3^{n-1}\)
Answer: \(29 \cdot 3^{n-1}\)
Question 27
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
25, 75, 225, 675, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 3\), \(a_1 = 25\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 25 \cdot 3^{n-1}\)
From the terms we find: \(q = 3\), \(a_1 = 25\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 25 \cdot 3^{n-1}\)
Answer: \(25 \cdot 3^{n-1}\)
Question 28
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
7, 35, 175, 875, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 5\), \(a_1 = 7\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 7 \cdot 5^{n-1}\)
From the terms we find: \(q = 5\), \(a_1 = 7\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 7 \cdot 5^{n-1}\)
Answer: \(7 \cdot 5^{n-1}\)
Question 29
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
21, 63, 189, 567, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 3\), \(a_1 = 21\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 21 \cdot 3^{n-1}\)
From the terms we find: \(q = 3\), \(a_1 = 21\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 21 \cdot 3^{n-1}\)
Answer: \(21 \cdot 3^{n-1}\)
Question 30
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
21, 84, 336, 1344, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 4\), \(a_1 = 21\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 21 \cdot 4^{n-1}\)
From the terms we find: \(q = 4\), \(a_1 = 21\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 21 \cdot 4^{n-1}\)
Answer: \(21 \cdot 4^{n-1}\)
Question 31
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
27, 135, 675, 3375, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 5\), \(a_1 = 27\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 27 \cdot 5^{n-1}\)
From the terms we find: \(q = 5\), \(a_1 = 27\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 27 \cdot 5^{n-1}\)
Answer: \(27 \cdot 5^{n-1}\)
Question 32
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
3, 6, 12, 24, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 2\), \(a_1 = 3\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 3 \cdot 2^{n-1}\)
From the terms we find: \(q = 2\), \(a_1 = 3\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 3 \cdot 2^{n-1}\)
Answer: \(3 \cdot 2^{n-1}\)
Question 33
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
8, 40, 200, 1000, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 5\), \(a_1 = 8\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 8 \cdot 5^{n-1}\)
From the terms we find: \(q = 5\), \(a_1 = 8\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 8 \cdot 5^{n-1}\)
Answer: \(8 \cdot 5^{n-1}\)
Question 34
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
9, 27, 81, 243, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 3\), \(a_1 = 9\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 9 \cdot 3^{n-1}\)
From the terms we find: \(q = 3\), \(a_1 = 9\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 9 \cdot 3^{n-1}\)
Answer: \(9 \cdot 3^{n-1}\)
Question 35
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
15, 60, 240, 960, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 4\), \(a_1 = 15\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 15 \cdot 4^{n-1}\)
From the terms we find: \(q = 4\), \(a_1 = 15\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 15 \cdot 4^{n-1}\)
Answer: \(15 \cdot 4^{n-1}\)
Question 36
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
22, 44, 88, 176, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 2\), \(a_1 = 22\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 22 \cdot 2^{n-1}\)
From the terms we find: \(q = 2\), \(a_1 = 22\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 22 \cdot 2^{n-1}\)
Answer: \(22 \cdot 2^{n-1}\)
Question 37
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
6, 24, 96, 384, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 4\), \(a_1 = 6\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 6 \cdot 4^{n-1}\)
From the terms we find: \(q = 4\), \(a_1 = 6\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 6 \cdot 4^{n-1}\)
Answer: \(6 \cdot 4^{n-1}\)
Question 38
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
2, 4, 8, 16, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 2\), \(a_1 = 2\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 2 \cdot 2^{n-1}\)
From the terms we find: \(q = 2\), \(a_1 = 2\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 2 \cdot 2^{n-1}\)
Answer: \(2 \cdot 2^{n-1}\)
Question 39
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
21, 105, 525, 2625, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 5\), \(a_1 = 21\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 21 \cdot 5^{n-1}\)
From the terms we find: \(q = 5\), \(a_1 = 21\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 21 \cdot 5^{n-1}\)
Answer: \(21 \cdot 5^{n-1}\)
Question 40
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
23, 69, 207, 621, ...
Find the general term formula \(a_n\).
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:
From the terms we find: \(q = 3\), \(a_1 = 23\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 23 \cdot 3^{n-1}\)
From the terms we find: \(q = 3\), \(a_1 = 23\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 23 \cdot 3^{n-1}\)
Answer: \(23 \cdot 3^{n-1}\)