solve the exponential sequence 2,4,8,16,32,64

The given sequence is:  
\[
2, 4, 8, 16, 32, 64
\]

This is an **exponential sequence** where each term is multiplied by a constant ratio to get the next term. Let's analyze it step-by-step.

### 1. **Identifying the pattern:**
Each term is obtained by multiplying the previous term by \(2\).

\[
\frac{4}{2} = 2, \quad \frac{8}{4} = 2, \quad \frac{16}{8} = 2, \quad \text{and so on...}
\]

This is a **geometric progression** with a **common ratio** \(r = 2\).

### 2. **General formula for a geometric sequence:**
The \(n\)-th term \(a_n\) of a geometric sequence is given by:
\[
a_n = a_1 \cdot r^{n - 1}
\]
where:
- \(a_1\) = first term = 2  
- \(r\) = common ratio = 2

### 3. **Formula for this sequence:**
\[
a_n = 2 \cdot 2^{n - 1} = 2^n
\]

### 4. **Verifying:**
- \(n = 1\), \(a_1 = 2^1 = 2\)
- \(n = 2\), \(a_2 = 2^2 = 4\)
- \(n = 3\), \(a_3 = 2^3 = 8\)
- \(n = 4\), \(a_4 = 2^4 = 16\)
- \(n = 5\), \(a_5 = 2^5 = 32\)
- \(n = 6\), \(a_6 = 2^6 = 64\)

The formula works perfectly!

### 5. **How to find any term:**
If you want to find any specific term in the sequence, just plug in the value of \(n\). For example, the 10th term \(a_{10}\) would be:
\[
a_{10} = 2^{10} = 1024
\]

Would you like to explore this further or ask anything else?

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### Related Questions:
1. What is the sum of the first \(n\) terms in a geometric sequence?
2. How can we determine the sum of an infinite geometric series?
3. How does a geometric sequence differ from an arithmetic sequence?
4. What are some real-world applications of geometric sequences?
5. Can a geometric sequence have a negative common ratio?

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**Tip:** Exponential growth increases rapidly—be careful with calculations involving large exponents to avoid errors!

Learn how to solve the exponential sequence 2, 4, 8, 16, 32, 64 by identifying the common ratio and using the geometric sequence formula. Explore exponential growth and geometric progressions with step-by-step explanations, suitable for students in Grades 6-8.

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