Question

Write the explicit formula for the sequence below and use it to find the 16th term:

Answer 1

Given:

The sequence is:

$6, -24, 96, -384,...$

To find:

The explicit formula for the given sequence and then find the 16th term.

Solution:

We have,

$6, -24, 96, -384,...$

The ratio between two consecutive terms are:

$\dfrac{-24}{6}=-4$

$\dfrac{96}{-24}=-4$

$\dfrac{-384}{96}=-4$

The given sequence has a common ratio. So, the given sequence is a geometric sequence with first term 6 and common ratio $-4$.

The explicit formula of a geometric sequence is:

$a_n=ar^{n-1}$

Where, a is the first term and r is the common ratio.

Putting $a=6, r=-4$ in the above formula, we get

$a_n=6(-4)^{n-1}$

We need to find the 16th term. So, put $n=16$ in the above formula.

$a_n=6(-4)^{16-1}$

$a_n=6(-4)^{15}$

$a_n=6(-1073741824)$

$a_n=-6442450944$

Therefore, the explicit formula for the given sequence is  $a_n=6(-4)^{n-1}$ and 16th term of the given sequence is $-6.442450944$.