Question

An infinite G.P has the first term $\text{'}x\text{'}$ and sum $\text{'}5\text{'}$ , then $x$ belongs to :

• A. $\left|x\right|<10$
• B. $-10<x<0$
• C. $0<x<10$
• D. $x>10$

Answer 1

The correct option is C

$0<x<10$

Determine the infinite G.P series:

The sum of Infinite GP $=ar+a{r}^2+a{r}^3+a{r}^4+....................$

Where,

• first term $=a$
• Common ratio $=r$

When $\left|r\right|<1$, the sum converges to $s_{\infty }=\frac{a}{1-r}$

When $\left|r\right|>1$, the series diverges because the total doesn't exist.

First-term $\left(a\right)=x$
Sum of infinite series $s_{\infty }=5$

Hence from the equation of $\left|r\right|<1$,

$$\begin{gathered} s_{\infty }=\frac{x}{1-r} \\ \Rightarrow 5=\frac{x}{1-r} \\ \Rightarrow 5-5r=x \\ \Rightarrow -r=\frac{x-5}{5} \\ \Rightarrow r=\frac{5-x}{5} \end{gathered}$$

But as$\left|r\right|<1$, so, the equation will be

$$\begin{gathered} -1<\frac{5-x}{5}<1 \\ \Rightarrow -5<5-x<5 \\ \Rightarrow -10<-x<0 \\ \Rightarrow 0<x<10 \end{gathered}$$

Hence, the correct choice is “Option C”.