Question

Find the 13th term of the geometric sequence 4, -16,64,

Answer 1

We are given the following geometric sequence

$4,-16,64,\ldots$

Let us first find a general formula for this sequence then we can easily find the 13th term.

Recall that a geometric sequence is given by

$a_n=a\cdot r^{n-1}$

Where aₙ is the nth term, a is the first term and r is the common ratio

The common ratio can be found by dividing the consecutive terms of the sequence.

$\begin{gathered} r=\frac{64}{-16}=-4 \\ r=-\frac{16}{4}=-4 \end{gathered}$

So the common ratio is -4 and the first term is 4

$a_n=4\cdot(-4)^{n-1}$

The above is the general formula for the sequence.

Now to find the 13th term, substitute n = 13 into the above formula

$\begin{gathered} a_{13}=4\cdot(-4)^{13-1} \\ a_{13}=4\cdot(-4)^{12} \\ a_{13}=4\cdot16777216 \\ a_{13}=67108864 \end{gathered}$

Therefore, the 13th term of the sequence is 67,108,864