Give me some geometric sequences problems and answers 

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

Give me some geometric sequences problems and answers

Explore detailed solutions to geometric sequence problems, including finding the n-th term, calculating sums, and determining common ratios. Perfect for high school students learning about sequences and series.

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