Question

Write an explicit formula for a, start subscript, n, end subscripta
n
​
, the n, start superscript, th, end superscriptn
th
term of the sequence 1, comma, 2, comma, 4, comma, dot dot dot1,2,4,....
Answer
Attempt 1 out of 2

a, start subscript, n, end subscript, equalsa
n
​
=

Answer 1

To find the explicit formula for the \( n \)-th term of the sequence \( 1, 2, 4, \ldots \), we need to identify the pattern.

The sequence can be expressed as:

- \( a_1 = 1 \)
- \( a_2 = 2 \)
- \( a_3 = 4 \)
- \( a_4 = 8 \)

We can observe that each term seems to be a power of 2:

- \( a_1 = 2^0 = 1 \)
- \( a_2 = 2^1 = 2 \)
- \( a_3 = 2^2 = 4 \)
- \( a_4 = 2^3 = 8 \)

Thus, in general, the \( n \)-th term can be expressed as:

\[
a_n = 2^{n-1}
\]

where \( n \) starts from 1.

So the explicit formula for \( a_n \) is:

\[
a_n = 2^{n-1}
\]