Question

The solution of the equation $1+\mathrm{a}+{\mathrm{a}}^2+{\mathrm{a}}^3+...{\mathrm{a}}^{\mathrm{x}}=(1+\mathrm{a})(1+{\mathrm{a}}^2)(1+{\mathrm{a}}^4)$. Then $\mathrm{x}$ is equal to

• A. $3$
• B. $5$
• C. $7$
• D. None of these

Answer 1

The correct option is C

$7$

The explanation for the correct answer.

Find the required value:

Given: $1+\mathrm{a}+{\mathrm{a}}^2+{\mathrm{a}}^3+...{\mathrm{a}}^{\mathrm{x}}=(1+\mathrm{a})(1+{\mathrm{a}}^2)(1+{\mathrm{a}}^4)$

The left-hand side of the equation is in GP.

The first term of the GP $=1$ and the common ratio $=\mathrm{a}$ and the last term $={\mathrm{a}}^{\mathrm{x}}$ .

The number of terms in the GP $=\mathrm{x}+1$.

Sum of the term of GP$=\frac{\mathrm{a}\left(1-{\mathrm{r}}^{\mathrm{n}}\right)}{1-\mathrm{r}}$, $\mathrm{a}$ is the first term and $\mathrm{r}$ is the common ratio.

$$\begin{gathered} \Rightarrow \frac{1\left(1-{\mathrm{a}}^{\mathrm{x}+1}\right)}{\left(1-\mathrm{a}\right)}=\left(1+\mathrm{a}\right)\left(1+{\mathrm{a}}^2\right)\left(1+{\mathrm{a}}^4\right)\left[\mathrm{Sum}\mathrm{is}\mathrm{given}\right] \\ \Rightarrow 1-{\mathrm{a}}^{\mathrm{x}+1}=\left(1-\mathrm{a}\right)\left(1+\mathrm{a}\right)\left(1+{\mathrm{a}}^2\right)\left(1+{\mathrm{a}}^4\right) \\ \Rightarrow 1-{\mathrm{a}}^{\mathrm{x}+1}=\left(1-{\mathrm{a}}^2\right)\left(1+{\mathrm{a}}^2\right)\left(1+{\mathrm{a}}^4\right)\left[\because \left(1-\mathrm{a}\right)\left(1+\mathrm{a}\right)=\left(1-{\mathrm{a}}^2\right)\right] \\ \Rightarrow 1-{\mathrm{a}}^{\mathrm{x}+1}=\left(1-{\mathrm{a}}^4\right)\left(1+{\mathrm{a}}^4\right)\left[\because \left(1-\mathrm{a}\right)\left(1+\mathrm{a}\right)=\left(1-{\mathrm{a}}^2\right)\right] \\ \Rightarrow 1-{\mathrm{a}}^{\mathrm{x}+1}=\left(1-{\mathrm{a}}^8\right)\left[\because \left(1-\mathrm{a}\right)\left(1+\mathrm{a}\right)=\left(1-{\mathrm{a}}^2\right)\right] \\ \Rightarrow {\mathrm{a}}^{\mathrm{x}+1}={\mathrm{a}}^8\left[\mathrm{Same}\mathrm{base},\mathrm{hence}\mathrm{indices}\mathrm{are}\mathrm{equal}\right] \\ \Rightarrow \mathrm{x}+1=8 \\ \Rightarrow \mathrm{x}=8-1 \\ \Rightarrow \mathrm{x}=7 \end{gathered}$$

Hence option $C$ is the correct answer.