Question

Write an equation to describe the sequence below.Use n to represent the position of a term in the sequence,where n = 1 for the first term.
1, 4, 16,
Write your answer using decimals and integers.
an = __ (__)n-1

Answer 1

To describe the sequence 1, 4, 16, we first recognize the pattern in the terms.

1 can be rewritten as \( 1^2 \) (which is 1),
4 can be rewritten as \( 2^2 \) (which is 4),
16 can be rewritten as \( 4^2 \) (which is 16).

More generally, we can observe that the terms are actually powers of 4:
- The first term (n=1) is \( 4^{(1-1)} = 4^0 = 1 \)
- The second term (n=2) is \( 4^{(2-1)} = 4^1 = 4 \)
- The third term (n=3) is \( 4^{(3-1)} = 4^2 = 16 \)

From this observation, we can derive the equation for the sequence:

\[
a_n = 4^{(n-1)}
\]

So, using the specified format:

\[
a_n = 4^{(n-1)}
\]