Question

How many terms are in the following geometric sequence? Type your numerical answer only. Do not type any additional characters.

Answer 1

Given:

The given geometric sequence is:

0.0625, 0.25, 1, ..., 4194304

To find:

The number of terms in the given geometric sequence.

Solution:

We have,

0.0625, 0.25, 1, ..., 4194304

Here, the first term is 0.0625 and the common ratio is:

$r=\dfrac{0.25}{0.0625}$

$r=4$

The nth term of a geometric sequence is:

$a_n=ar^{n-1}$

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

Putting $a_n=4194304, a=0.0625, r=4$ in the above formula, we get

$4194304=0.0625(4)^{n-1}$

$\dfrac{4194304}{0.0625}=(4)^{n-1}$

$67108864=(4)^{n-1}$

$4^{13}=(4)^{n-1}$

On comparing both sides, we get

$13=n-1$

$13+1=n$

$14=n$

Therefore, the number of terms in the given geometric sequence is 14.