Question

What is the explicit formula for the geometric sequence with this recursive formula?

{a(1) = -3
{a(n) = a(n-1) • (1/8)

Answer 1

To find the explicit formula for the geometric sequence defined by the recursive formula

\[
a(1) = -3
\]
\[
a(n) = a(n-1) \cdot \left(\frac{1}{8}\right)
\]

we first note the characteristics of a geometric sequence. A geometric sequence has a common ratio, which in this case is \( r = \frac{1}{8} \).

The first term of the sequence is \( a(1) = -3 \).

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

\[
a(n) = a(1) \cdot r^{n-1}
\]

Substituting in the first term and the common ratio, we have:

\[
a(n) = -3 \cdot \left(\frac{1}{8}\right)^{n-1}
\]

Thus, the explicit formula for the sequence is:

\[
\boxed{a(n) = -3 \cdot \left(\frac{1}{8}\right)^{n-1}}
\]