Question

What is the explicit formula for the geometric sequence with this recursive formula?
{a1 = -7
{an = an - 1 • (1/3)

Answer 1

To find the explicit formula for the geometric sequence given the recursive formula:

\[
a_1 = -7
\]
\[
a_n = a_{n-1} \cdot \frac{1}{3}
\]

we can recognize that this is a geometric sequence where the first term \( a_1 \) is \(-7\) and the common ratio \( r \) is \(\frac{1}{3}\).

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

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

Substituting the values we have:

- \( a_1 = -7 \)
- \( r = \frac{1}{3} \)

The explicit formula becomes:

\[
a_n = -7 \cdot \left(\frac{1}{3}\right)^{n-1}
\]

Thus, the explicit formula for the sequence is:

\[
a_n = -7 \cdot \left(\frac{1}{3}\right)^{n-1}
\]

This formula can be used to calculate any term in the sequence.