Question

Which term of the G.P. $2,8,32,...$ is $512$?

Answer 1

First we will find the first term and common ratio of the given G.P. Then we assume the term we have to find to be the nth term. Now, we can use the formula of nth term which gives the relation between the first term, common ratio, and nth term of the G.P and is given as-
$\Rightarrow {{\text{T}}_{\text{n}}} = a{r^{n - 1}}$
Here, ‘a’ is the first term of the series, r is the common ratio and n is the nth term of the series. Put the given values and solve the required equation.

Complete step-by-step answer:
Here, the given series is $2,8,32,...$ which is in G.P.
We have to find the term whose value is $512$ in this series.
Here, the first term ‘a’=$2$
Common ratio r=$\dfrac{8}{2} = \dfrac{{32}}{8} = 4$
Let the ${{\text{n}}^{{\text{th}}}}$ term be $512$
Now we only need to find n.
We know the formula of ${{\text{n}}^{{\text{th}}}}$ term is given as-
$\Rightarrow {{\text{T}}_{\text{n}}} = a{r^{n - 1}}$
Here, ‘a’ is the first term of the series, r is the common ratio and n is the nth term of the series.
On putting, the given values in the above formula, we get-
$\Rightarrow {\text{512}} = 2{\left( 4 \right)^{n - 1}}$
On solving, we get-
$\Rightarrow \dfrac{{{\text{512}}}}{2} = {\left( 4 \right)^{n - 1}}$
On dividing the numerator by denominator on the left side, we get-
$\Rightarrow 256 = {\left( 4 \right)^{n - 1}}$
Now we will find the factors of $256$ so that we can write it in terms of base $4$ so we can write $256 = 4 \times 4 \times 4 \times 4 = {4^4}$ then, we get-
$\Rightarrow {4^4} = {\left( 4 \right)^{n - 1}}$
Since here the base on both sides is the same so the power of the base should also be equal on both sides. So on comparing powers, we get-
$\Rightarrow 4 = n - 1$
On solving, we get-
$\Rightarrow n = 4 + 1$
On further solving, we get-
$\Rightarrow n = 5$
So here the nth term is ${5^{{\text{th}}}}$ term.
Hence, $512$ is ${5^{{\text{th}}}}$ term of the given G.P. series.

Note: In this type of question, we assume the value of the term we have to find to be the nth term so that we can easily find the value of n using the formula of the nth term in the G.P. series. We can also find terms using this formula provided that we know the value of a, r and n.