find the nth term of 2, 4, 8, 16, 32 and show solution

To find the \( n \)th term of the sequence \( 2, 4, 8, 16, 32, \ldots \), we need to recognize the pattern and identify the type of sequence.

The given sequence is \( 2, 4, 8, 16, 32, \ldots \).

Observing the pattern:
- \( 4 = 2 \times 2 \)
- \( 8 = 4 \times 2 \)
- \( 16 = 8 \times 2 \)
- \( 32 = 16 \times 2 \)

Each term is obtained by multiplying the previous term by 2. Therefore, this sequence is a geometric sequence where the first term \( a = 2 \) and the common ratio \( r = 2 \).

The general formula for the \( n \)th term of a geometric sequence is:
\[ a_n = a \cdot r^{n-1} \]

Here:
- \( a = 2 \)
- \( r = 2 \)

Thus, the \( n \)th term \( a_n \) is:
\[ a_n = 2 \cdot 2^{n-1} \]

### Solution:
To find the \( n \)th term of the sequence \( 2, 4, 8, 16, 32, \ldots \), use the formula \( a_n = 2 \cdot 2^{n-1} \).

Let's verify this formula with a few terms of the sequence:

- For \( n = 1 \):
  \[ a_1 = 2 \cdot 2^{1-1} = 2 \cdot 2^0 = 2 \cdot 1 = 2 \]

- For \( n = 2 \):
  \[ a_2 = 2 \cdot 2^{2-1} = 2 \cdot 2^1 = 2 \cdot 2 = 4 \]

- For \( n = 3 \):
  \[ a_3 = 2 \cdot 2^{3-1} = 2 \cdot 2^2 = 2 \cdot 4 = 8 \]

- For \( n = 4 \):
  \[ a_4 = 2 \cdot 2^{4-1} = 2 \cdot 2^3 = 2 \cdot 8 = 16 \]

- For \( n = 5 \):
  \[ a_5 = 2 \cdot 2^{5-1} = 2 \cdot 2^4 = 2 \cdot 16 = 32 \]

This confirms that the formula \( a_n = 2 \cdot 2^{n-1} \) correctly represents the \( n \)th term of the given sequence.

Would you like more details or have any questions?

### Relative Questions:

1. What is a geometric sequence and how is it different from an arithmetic sequence?
2. Can you derive the formula for the \( n \)th term of a geometric sequence?
3. What is the sum of the first \( n \) terms of a geometric sequence?
4. How do you find the common ratio in a geometric sequence?
5. What is the formula for the \( n \)th term if the first term and common ratio are given?
6. How would you determine the number of terms in a geometric sequence if the first term, common ratio, and last term are known?
7. Can you provide a real-world example where a geometric sequence is applicable?
8. How does changing the common ratio affect the sequence?

### Tip:
When working with sequences, always identify whether it is arithmetic, geometric, or another type. This identification helps in applying the correct formulas for finding terms and sums.

Discover how to find the nth term of the geometric sequence 2, 4, 8, 16, 32 using the geometric sequence nth term formula. This problem explores geometric sequences, their properties, and specific calculations. Suitable for students in Grades 10-12.

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