Chapter 12.3, Problem 42E

To determine

Find the number of term of $31250$ for the given second term $a_2=10$ & fifth term $a_5=1250$ of the geometric sequence

Answer to Problem 42E

The number of term of $118098$ for the given second term $a_2=10$ & fifth term $a_5=1250$ of the geometric sequence is $n=7th$ .

Explanation of Solution

Given information:

It is given in the question that, second term $a_2=10$ & fifth term $a_5=1250$ of the geometric sequence.

Formula used:

  $a_n=a{r}^{n-1}$

Where,

  $\begin{array}{l}aisthe{1}^{st}term\\ nisthenumberofterm\\ risthecommonratio\end{array}$

Calculation:

Second term $a_2=10$ & fifth term $a_5=1250$ of the geometric sequence means,

  $\begin{array}{l}{a}_2=10\Rightarrow ar=10\Rightarrow a=\frac{10}{r}\\ a_5=1250\Rightarrow a{r}^4=1250\Rightarrow \frac{10}{r}*r^4=1250\Rightarrow 10*r^3=1250\Rightarrow r^3=125\Rightarrow r=5\\ byputtingr=5ina=\frac{10}{r}\Rightarrow a=\frac{10}{5}\Rightarrow a=2\end{array}$

From above,

  $a=2\&r=5$

Let the given number $31250$ be $a_n$ so $a_n=31250$

By using the formula $a_n=a{r}^{n-1}$

  $\begin{array}{l}{a}_n=31250\\ a{r}^{n-1}=31250\\ 2*5^{n-1}=31250\\ 5^{n-1}=\frac{31250}{2}\\ 5^{n-1}=15625\\ 5^{n-1}=5^6\end{array}$

So by equating, both the above term,

  $\begin{array}{l}n-1=6\\ n=7\end{array}$