Question

If the first term and common ratio of a G.P. are 5 and -5 respectively, then the ___ term is 3125.

• A. $6^{\text{th}}$
• B. $5^{\text{th}}$
• C. $7^{\text{th}}$
• D. $8^{\text{th}}$

Answer 1

The correct option is B

$5^{\text{th}}$

Given that the first term $a=5$ and common ratio $r=-5.$

Suppose that $n^{\text{th}}$ term is 3125.

$\Rightarrow a{r}^{n-1}=3125$
$\Rightarrow 5(-5{)}^{n-1}=3125$
$\Rightarrow (-5{)}^{n-1}=5^4$
i.e., $(-1{)}^{n-1}\times (5{)}^{n-1}=5^4$
Since $\frac{5^4}{5^{n-1}}>0,$ we must have $(-1{)}^{n-1}=1.$
$\Rightarrow 5^{n-1}=4\Rightarrow n-1=4\Rightarrow n=5$
Thus, the $5^{\text{th}}$ term of the GP is 3125.