What is the explicit formula for the sequence 4 8 16 32?

The explicit formula for the sequence 4, 8, 16, 32 is a_n = 4(2)^n-1. This sequence is a geometric sequence.

Understanding Geometric Sequences

A geometric sequence is a sequence where the ratio between any two consecutive terms remains constant. This constant ratio is called the common ratio. In the sequence 4, 8, 16, 32, the common ratio is 2 (8/4 = 2, 16/8 = 2, 32/16 = 2).

Finding the Explicit Formula

The general formula for the nth term (a_n) of a geometric sequence is:

a_n = a₁ * r^n-1

Where:

• a₁ is the first term of the sequence.
• r is the common ratio.
• n is the term number (e.g., 1st term, 2nd term, etc.).

In our case:

• a₁ = 4
• r = 2

Therefore, the explicit formula for the sequence 4, 8, 16, 32 is:

a_n = 4 * (2)^n-1

Example Usage

Let's find the 4th term of the sequence using the formula:

a₄ = 4 (2)^4-1
a₄ = 4 (2)³
a₄ = 4 * 8
a₄ = 32

This confirms that the formula correctly calculates the terms of the sequence.

Summary

Property	Value
First Term (a1)	4
Common Ratio (r)	2
Explicit Formula	an = 4(2)n-1