Geometric Sequence — General Term Formula — Dynamic Practice
Geometric Sequence — General Term Formula — Dynamic Practice. Practice questions to deepen understanding of finding and applying the general term formula of a geometric sequence. Online math practice with full solutions and detailed explanations.
Dynamic practice in deriving the formula for aₙ in a geometric sequence — using aₙ = a₁ · q^(n−1) from given data. 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:
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 2
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
8, 32, 128, 512, ...
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: \(q = 4\), \(a_1 = 8\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 8 \cdot 4^{n-1}\)
From the terms we find: \(q = 4\), \(a_1 = 8\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 8 \cdot 4^{n-1}\)
Answer: \(8 \cdot 4^{n-1}\)
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:
19, 57, 171, 513, ...
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: \(q = 3\), \(a_1 = 19\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 19 \cdot 3^{n-1}\)
From the terms we find: \(q = 3\), \(a_1 = 19\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 19 \cdot 3^{n-1}\)
Answer: \(19 \cdot 3^{n-1}\)
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:
10, 30, 90, 270, ...
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: \(q = 3\), \(a_1 = 10\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 10 \cdot 3^{n-1}\)
From the terms we find: \(q = 3\), \(a_1 = 10\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 10 \cdot 3^{n-1}\)
Answer: \(10 \cdot 3^{n-1}\)
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:
22, 110, 550, 2750, ...
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: \(q = 5\), \(a_1 = 22\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 22 \cdot 5^{n-1}\)
From the terms we find: \(q = 5\), \(a_1 = 22\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 22 \cdot 5^{n-1}\)
Answer: \(22 \cdot 5^{n-1}\)
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:
25, 50, 100, 200, ...
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: \(q = 2\), \(a_1 = 25\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 25 \cdot 2^{n-1}\)
From the terms we find: \(q = 2\), \(a_1 = 25\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 25 \cdot 2^{n-1}\)
Answer: \(25 \cdot 2^{n-1}\)
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:
6, 30, 150, 750, ...
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: \(q = 5\), \(a_1 = 6\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 6 \cdot 5^{n-1}\)
From the terms we find: \(q = 5\), \(a_1 = 6\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 6 \cdot 5^{n-1}\)
Answer: \(6 \cdot 5^{n-1}\)
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:
25, 125, 625, 3125, ...
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: \(q = 5\), \(a_1 = 25\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 25 \cdot 5^{n-1}\)
From the terms we find: \(q = 5\), \(a_1 = 25\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 25 \cdot 5^{n-1}\)
Answer: \(25 \cdot 5^{n-1}\)
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:
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 10
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
17, 85, 425, 2125, ...
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: \(q = 5\), \(a_1 = 17\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 17 \cdot 5^{n-1}\)
From the terms we find: \(q = 5\), \(a_1 = 17\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 17 \cdot 5^{n-1}\)
Answer: \(17 \cdot 5^{n-1}\)
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:
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 12
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
15, 30, 60, 120, ...
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: \(q = 2\), \(a_1 = 15\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 15 \cdot 2^{n-1}\)
From the terms we find: \(q = 2\), \(a_1 = 15\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 15 \cdot 2^{n-1}\)
Answer: \(15 \cdot 2^{n-1}\)
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:
10, 40, 160, 640, ...
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: \(q = 4\), \(a_1 = 10\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 10 \cdot 4^{n-1}\)
From the terms we find: \(q = 4\), \(a_1 = 10\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 10 \cdot 4^{n-1}\)
Answer: \(10 \cdot 4^{n-1}\)
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:
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 15
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
11, 55, 275, 1375, ...
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: \(q = 5\), \(a_1 = 11\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 11 \cdot 5^{n-1}\)
From the terms we find: \(q = 5\), \(a_1 = 11\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 11 \cdot 5^{n-1}\)
Answer: \(11 \cdot 5^{n-1}\)
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:
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 17
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 18
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 19
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
12, 36, 108, 324, ...
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: \(q = 3\), \(a_1 = 12\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 12 \cdot 3^{n-1}\)
From the terms we find: \(q = 3\), \(a_1 = 12\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 12 \cdot 3^{n-1}\)
Answer: \(12 \cdot 3^{n-1}\)
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:
27, 54, 108, 216, ...
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: \(q = 2\), \(a_1 = 27\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 27 \cdot 2^{n-1}\)
From the terms we find: \(q = 2\), \(a_1 = 27\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 27 \cdot 2^{n-1}\)
Answer: \(27 \cdot 2^{n-1}\)
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:
9, 18, 36, 72, ...
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: \(q = 2\), \(a_1 = 9\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 9 \cdot 2^{n-1}\)
From the terms we find: \(q = 2\), \(a_1 = 9\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 9 \cdot 2^{n-1}\)
Answer: \(9 \cdot 2^{n-1}\)
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:
29, 116, 464, 1856, ...
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: \(q = 4\), \(a_1 = 29\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 29 \cdot 4^{n-1}\)
From the terms we find: \(q = 4\), \(a_1 = 29\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 29 \cdot 4^{n-1}\)
Answer: \(29 \cdot 4^{n-1}\)
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:
3, 12, 48, 192, ...
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: \(q = 4\), \(a_1 = 3\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 3 \cdot 4^{n-1}\)
From the terms we find: \(q = 4\), \(a_1 = 3\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 3 \cdot 4^{n-1}\)
Answer: \(3 \cdot 4^{n-1}\)
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:
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 25
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 26
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 27
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
27, 81, 243, 729, ...
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: \(q = 3\), \(a_1 = 27\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 27 \cdot 3^{n-1}\)
From the terms we find: \(q = 3\), \(a_1 = 27\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 27 \cdot 3^{n-1}\)
Answer: \(27 \cdot 3^{n-1}\)
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:
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 29
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
30, 60, 120, 240, ...
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: \(q = 2\), \(a_1 = 30\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 30 \cdot 2^{n-1}\)
From the terms we find: \(q = 2\), \(a_1 = 30\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 30 \cdot 2^{n-1}\)
Answer: \(30 \cdot 2^{n-1}\)
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:
6, 12, 24, 48, ...
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: \(q = 2\), \(a_1 = 6\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 6 \cdot 2^{n-1}\)
From the terms we find: \(q = 2\), \(a_1 = 6\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 6 \cdot 2^{n-1}\)
Answer: \(6 \cdot 2^{n-1}\)
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:
29, 145, 725, 3625, ...
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: \(q = 5\), \(a_1 = 29\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 29 \cdot 5^{n-1}\)
From the terms we find: \(q = 5\), \(a_1 = 29\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 29 \cdot 5^{n-1}\)
Answer: \(29 \cdot 5^{n-1}\)
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:
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 33
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
17, 34, 68, 136, ...
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: \(q = 2\), \(a_1 = 17\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 17 \cdot 2^{n-1}\)
From the terms we find: \(q = 2\), \(a_1 = 17\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 17 \cdot 2^{n-1}\)
Answer: \(17 \cdot 2^{n-1}\)
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:
12, 48, 192, 768, ...
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: \(q = 4\), \(a_1 = 12\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 12 \cdot 4^{n-1}\)
From the terms we find: \(q = 4\), \(a_1 = 12\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 12 \cdot 4^{n-1}\)
Answer: \(12 \cdot 4^{n-1}\)
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:
11, 33, 99, 297, ...
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: \(q = 3\), \(a_1 = 11\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 11 \cdot 3^{n-1}\)
From the terms we find: \(q = 3\), \(a_1 = 11\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 11 \cdot 3^{n-1}\)
Answer: \(11 \cdot 3^{n-1}\)
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:
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 37
2.50 pts
📊 Geometric Sequence:
Given a geometric sequence whose first four terms are:
Given a geometric sequence whose first four terms are:
28, 112, 448, 1792, ...
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: \(q = 4\), \(a_1 = 28\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 28 \cdot 4^{n-1}\)
From the terms we find: \(q = 4\), \(a_1 = 28\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 28 \cdot 4^{n-1}\)
Answer: \(28 \cdot 4^{n-1}\)
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:
22, 88, 352, 1408, ...
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: \(q = 4\), \(a_1 = 22\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 22 \cdot 4^{n-1}\)
From the terms we find: \(q = 4\), \(a_1 = 22\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 22 \cdot 4^{n-1}\)
Answer: \(22 \cdot 4^{n-1}\)
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:
28, 56, 112, 224, ...
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: \(q = 2\), \(a_1 = 28\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 28 \cdot 2^{n-1}\)
From the terms we find: \(q = 2\), \(a_1 = 28\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 28 \cdot 2^{n-1}\)
Answer: \(28 \cdot 2^{n-1}\)
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:
25, 100, 400, 1600, ...
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: \(q = 4\), \(a_1 = 25\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 25 \cdot 4^{n-1}\)
From the terms we find: \(q = 4\), \(a_1 = 25\)
The formula: \(a_n = a_1 \cdot q^{n-1} = 25 \cdot 4^{n-1}\)
Answer: \(25 \cdot 4^{n-1}\)