Question

There are two series given $1,4,16,64,\cdots$ and $1,5,25,125,\cdots$ of $10$ terms each. If both the series are in $\text{G.P.}$ then find the number of trailing zeros in the $9^{th}$ term obtained after multiplying both the series.

Answer 1

Given: $1,4,16,64,\cdots$ and $1,5,25,125,\cdots$ of $10$ terms each are in $\text{G.P.}$

To Find: Number of trailing zeros in the $9^{th}$ term obtained after multiplying both the series.

Series: $1,4,16,64,\cdots \&1,5,25,125,\cdots$

On multiplying both the series, we get $1,20,400,\cdots$

We know that, multiplication /division of respective terms of two $\text{G.P.s}$ also forms a $\text{G.P.}$

Hence, $1,20,400,\cdots$ is in $\text{G.P.},$ with first term $a=1$ and common ratio $r=20.$

The $9^{th}$ term will be $a_9=a{r}^8=1\times {20}^8$

$a_9=25600000000$

There are $8$ zeros.