Question

If ${\left(10\right)}^9+2{\left(11\right)}^1{\left(10\right)}^8+3{\left(11\right)}^2{\left(10\right)}^7+.........+10{\left(11\right)}^9=k{\left(10\right)}^9,$ then $k$ is equal to

• A. $100$
• B. $110$
• C. $\frac{121}{100}$
• D. $\frac{441}{100}$

Answer 1

The correct option is A

$100$

Explanation for correct option:

Step-1: Simplify the given data.

Given, ${\left(10\right)}^9+2{\left(11\right)}^1{\left(10\right)}^8+3{\left(11\right)}^2{\left(10\right)}^7+.........+10{\left(11\right)}^9=k{\left(10\right)}^9,$

$\Rightarrow$ ${\left(10\right)}^9\left(1+2\frac{{\left(11\right)}^1}{10}+3\frac{{\left(11\right)}^2}{{\left(10\right)}^2}+.........+10\frac{{\left(11\right)}^9}{{\left(10\right)}^9}\right)=k{\left(10\right)}^9,$

$\Rightarrow$ $\left(1+2\frac{{\left(11\right)}^1}{10}+3\frac{{\left(11\right)}^2}{{\left(10\right)}^2}+.........+10\frac{{\left(11\right)}^9}{{\left(10\right)}^9}\right)=k,......\left(i\right)$

Multiply by $\left(\frac{11}{10}\right)$ both the sides.

$\Rightarrow$ $\left(\frac{11}{10}+2\frac{{\left(11\right)}^2}{{\left(10\right)}^2}+3\frac{{\left(11\right)}^3}{{\left(10\right)}^3}+.........+10\frac{{\left(11\right)}^{10}}{{\left(10\right)}^{10}}\right)=\left(\frac{11}{10}\right)k,.........\left(ii\right)$

Step-2: Subtract equation $\left(ii\right)$ from equation $\left(i\right)$.

$\Rightarrow$ $\left(1+\frac{11}{10}+\frac{{\left(11\right)}^2}{{\left(10\right)}^2}+\frac{{\left(11\right)}^3}{{\left(10\right)}^3}+.........-10\frac{{\left(11\right)}^{10}}{{\left(10\right)}^{10}}\right)=\left(1-\frac{11}{10}\right)k,$

$\Rightarrow$$\left(\left(1+\frac{11}{10}+\frac{{\left(11\right)}^2}{{\left(10\right)}^2}+.........+\frac{{\left(11\right)}^9}{{\left(10\right)}^9}\right)-10\frac{{\left(11\right)}^{10}}{{\left(10\right)}^{10}}\right)=-\frac{k}{10},$

$\Rightarrow$ $\left(1\left(\frac{{\left({\displaystyle \frac{11}{10}}\right)}^{10}-1}{\left({\displaystyle \frac{11}{10}-1}\right)}\right)-10\frac{{\left(11\right)}^{10}}{{\left(10\right)}^{10}}\right)=-\frac{k}{10},$ ; $\left[\left(\because S_n=\frac{a\left(r^n-1\right)}{r-1}\right)\right]$

$\Rightarrow$ $10\left(\left(\frac{{\left({\displaystyle \frac{11}{10}}\right)}^{10}-1}{\left({\displaystyle \frac{1}{10}}\right)}\right)-10\frac{{\left(11\right)}^{10}}{{\left(10\right)}^{10}}\right)=-k,$

$\Rightarrow$ $10\left(10{\left(\frac{11}{10}\right)}^{10}-10-10\frac{{\left(11\right)}^{10}}{{\left(10\right)}^{10}}\right)=-k,$

$\Rightarrow$ $-100=-k,$

$\Rightarrow$ $k=100,$

Hence, correct answer is option $\left(A\right).$