Here are five examples of geometric sequence problems, each followed by a detailed solution:

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1. Find the 7th term of a geometric sequence

Problem: A geometric sequence has a first term $a_1 = 3$ and a common ratio $r = 2$. Find the 7th term.

Solution:
The general formula for the $n$-th term of a geometric sequence is: $a_n = a_1 \cdot r^{n-1}$

Substitute $a_1 = 3$, $r = 2$, and $n = 7$: $a_7 = 3 \cdot 2^{7-1} = 3 \cdot 2^6 = 3 \cdot 64 = 192$

Answer: The 7th term is $192$.

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2. Sum of the first 5 terms of a geometric sequence

Problem: Find the sum of the first 5 terms of a geometric sequence where $a_1 = 4$ and $r = 3$.

Solution:
The sum of the first $n$ terms of a geometric sequence is given by: $S_n = a_1 \cdot \frac{1 - r^n}{1 - r}, \quad \text{if } r \neq 1$

Substitute $a_1 = 4$, $r = 3$, and $n = 5$: $S_5 = 4 \cdot \frac{1 - 3^5}{1 - 3}$ $S_5 = 4 \cdot \frac{1 - 243}{-2} = 4 \cdot \frac{-242}{-2} = 4 \cdot 121 = 484$

Answer: The sum of the first 5 terms is $484$.

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3. Find the common ratio of a geometric sequence

Problem: The first term of a geometric sequence is $5$, and the third term is $45$. Find the common ratio.

Solution:
The general formula for the $n$-th