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 8, comma, 4, comma, 2, comma, dot dot dot8,4,2,....

Answer 1

To derive an explicit formula for the sequence \( a_n \), where the terms are \( 8, 4, 2, \ldots \), we can observe the following:

1. The first term \( a_1 = 8 \).
2. The second term \( a_2 = 4 \).
3. The third term \( a_3 = 2 \).

Each term is getting halved compared to the previous term. Thus, we can express each term in terms of the previous one:

\[
a_n = \frac{a_{n-1}}{2}
\]

This indicates a geometric sequence, where the first term is \( a_1 = 8 \) and the common ratio \( r = \frac{1}{2} \).

The general formula for the \( n \)-th term of a geometric sequence is given by:

\[
a_n = a_1 \cdot r^{n-1}
\]

Plugging in the values we have:

- \( a_1 = 8 \)
- \( r = \frac{1}{2} \)

So we obtain:

\[
a_n = 8 \cdot \left( \frac{1}{2} \right)^{n-1}
\]

This can also be rewritten as:

\[
a_n = \frac{8}{2^{n-1}}
\]

Thus, the explicit formula for the \( n \)-th term of the sequence is:

\[
a_n = \frac{8}{2^{n-1}}
\]