Question

If the nth term of a GP $5,-5/2,5/4,-5/8,.....$ is $\frac{5}{1024}$, then the value of $n$ is

• A. $11$
• B. $10$
• C. $9$
• D. $4$

Answer 1

The correct option is A

$11$

Explanation for the correct answer.

Given: GP $5,-5/2,5/4,-5/8,.....$ and nth term of GP is $\frac{5}{1024}$

Let $a$be the first term and $r$ be the common ratio of the given GP

$$\begin{gathered} a=5 \\ r=\frac{a_2}{a_1}=\frac{{\displaystyle \raisebox{1ex}{$-5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{5}=\frac{-1}{2} \end{gathered}$$

Nth term of GP is $a_n=a{r}^{n-1}$

$\begin{array}{cccc}\Rightarrow & a{r}^{n-1}& =& \frac{5}{1024}\\ \Rightarrow & 5{\left(\frac{-1}{2}\right)}^{n-1}& =& \frac{5}{1024}\\ \Rightarrow & {\left(\frac{-1}{2}\right)}^{n-1}& =& {\left(-\frac{1}{2}\right)}^{10}\\ \Rightarrow & n-1& =& 10\\ \Rightarrow & n& =& 11\end{array}$

Hence, option (a) is correct.