Question

Which term of the G.P. $2,2\sqrt{2},4,.......$ is 32?

Answer 1

Hint:We have to know about G.P i.e. Geometric Progression. It is a sequence in which any element after the first element can be obtained by multiplying the preceding element by a constant known as common ratio. If we have a G.P ${{a}_{1}},{{a}_{2}},{{a}_{3}}....{{a}_{n}}$ then
$\begin{align}
  & common\ ratio=r=\dfrac{{{a}_{2}}}{{{a}_{1}}}=\dfrac{{{a}_{3}}}{{{a}_{2}}}.....\dfrac{{{a}_{n}}}{{{a}_{n-1}}} \\
 & {{n}^{th}}term={{T}_{n}}={{a}_{1}}{{r}^{n-1}} \\
\end{align}$
Here, a, is the first term and r is the common ratio.

Complete step-by-step answer:
We have been asked to find which term of the G.P. $2,2\sqrt{2},4,.......$ is 32.
We know that if we have a G.P ${{a}_{1}},{{a}_{2}},{{a}_{3}}....{{a}_{n}}$ then,
$common\ ratio=r=\dfrac{{{a}_{2}}}{{{a}_{1}}}=\dfrac{{{a}_{3}}}{{{a}_{2}}}.....$
We have G.P. $2,2\sqrt{2},4,.......$
$common\ ratio\left( r \right)=\dfrac{2\sqrt{2}}{2}=\sqrt{2}$
Let us suppose the \[{{n}^{th}}\] term of the given G.P is 32.
We know that the \[{{n}^{th}}\] term of any G.P is given by,
${{T}_{n}}=a{{r}^{n-1}}$
Where a is the first term and r is the common ratio.
$\Rightarrow 32=2\times {{\left( \sqrt{2} \right)}^{n-1}}$
We know that 32 can be written in the form of power of 2 as ${{2}^{5}}$.
$\Rightarrow {{2}^{5}}=2\times {{\left( \sqrt{2} \right)}^{n-1}}$
We know $\sqrt{2}={{\left( 2 \right)}^{\dfrac{1}{2}}}$
$\Rightarrow {{2}^{5}}=2\times {{2}^{\dfrac{n-1}{2}}}$
We know that ${{a}^{{{x}_{1}}}}\times {{a}^{{{x}_{2}}}}={{a}^{{{x}_{1}}+{{x}_{2}}}}$.
$\begin{align}
  & \Rightarrow {{2}^{5}}={{2}^{1}}\times {{2}^{\dfrac{n-1}{2}}} \\
 & \Rightarrow {{2}^{5}}={{2}^{1+\dfrac{n-1}{2}}} \\
 & \Rightarrow {{2}^{5}}={{2}^{\dfrac{2+n-1}{2}}} \\
 & \Rightarrow {{2}^{5}}={{2}^{\dfrac{n+1}{2}}} \\
\end{align}$
On comparing both sides we get the base are same which means the power must be same to hold the equality.
$\begin{align}
  & \Rightarrow \dfrac{n+1}{2}=5 \\
 & \Rightarrow n+1=10 \\
 & \Rightarrow n=10-1 \\
 & \Rightarrow n=9 \\
\end{align}$
Therefore, the 9th term of the given G.P is 32.

Note: Remember the point that while solving for the value of ‘n’ if the value of ‘n’ is a whole number then your answer will be correct but if you get a fraction i.e. not a whole number then you must check your calculation or it is possible that the given term doesn’t come in the given G.P.