Question

Which formula can be used to find the nth term of a geometric sequence where the fifth term is 1/16 and the common ratio is 1/4?

Answer 1

To find the nth term of a geometric sequence, you can use the formula:

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

where:
- \(a_n\) is the nth term,
- \(a_1\) is the first term,
- \(r\) is the common ratio,
- \(n\) is the term number.

In this case, you are given that the fifth term (\(a_5\)) is \(1/16\) and the common ratio \(r\) is \(1/4\).

Using the formula for the fifth term:

\[
a_5 = a_1 \cdot r^{(5-1)} = a_1 \cdot r^4
\]

Substituting \(r = \frac{1}{4}\):

\[
a_5 = a_1 \cdot \left(\frac{1}{4}\right)^4
\]
\[
= a_1 \cdot \frac{1}{256}
\]

We know that \(a_5 = \frac{1}{16}\). Therefore, we can set up the following equation:

\[
a_1 \cdot \frac{1}{256} = \frac{1}{16}
\]

To solve for \(a_1\), multiply both sides by \(256\):

\[
a_1 = \frac{1}{16} \cdot 256
\]
\[
= 16
\]

Now, we have \(a_1 = 16\) and \(r = \frac{1}{4}\).

Now we can express the nth term of the geometric sequence:

\[
a_n = 16 \cdot \left(\frac{1}{4}\right)^{(n-1)}
\]

Thus, the formula to find the nth term of this geometric sequence is:

\[
a_n = 16 \cdot \left(\frac{1}{4}\right)^{(n-1)}
\]