Question

• Answer the question without spamming :- If (10)⁹ + 2(11)¹(10)⁸ + 3(11)²(10)⁷ + ... + 10(11)⁹ = k(10)⁹, then find the value of k.

Answer 1

◙ Given ◙

(10)⁹ + 2(11)¹(10)⁸ + 3(11)²(10)⁷ + ... + 10(11)⁹ = k(10)⁹.

◙ To Find ◙

The value of k.

◙ Solution ◙

We have :-

$\sf{(10)^9 + 2(11)^1(10)^8 + 3(11)^2(10)^7 + ... + 10(11)^9 = k(10)^9}$

Dividing (10)⁹ on both the sides :-

$\sf{\longrightarrow 1+2\bigg(\dfrac{11}{10} \bigg)+3 \bigg(\dfrac{11}{10} \bigg) ^2+...+10 \bigg(\dfrac{11}{10} \bigg) ^9 =k}\:\:\:\: \cdots (i)$

Multiplying 11/10 on both the sides of (i) :-

$\longrightarrow \sf{ \bigg( \dfrac{11}{10} \bigg)+2\bigg(\dfrac{11}{10} \bigg)^2+3\bigg(\dfrac{11}{10} \bigg)^2 +...+10\bigg(\dfrac{11}{10} \bigg)^{10}=\bigg(\dfrac{11}{10} \bigg)k }\:\:\:\: \cdots (ii)$

Subtracting (ii) from (i), we get :-

$\longrightarrow \displaystyle \sf{k- \bigg(\frac{11}{10}\bigg)k=1+\bigg(\frac{11}{10}\bigg)+\bigg(\frac{11}{10}\bigg)^2+...+\bigg(\frac{11}{10}\bigg)^9-10\bigg(\frac{11}{10}\bigg)^{10} }$

$\displaystyle \sf{\longrightarrow k\bigg[1-\bigg(\frac{11}{10} \bigg) \bigg]=\frac{1\big[\big(\frac{11}{10} \big)^{10}-1 \big]}{\big(\frac{11}{10} \big)-1} -10\bigg( \frac{11}{10} \bigg)^{10}}$

$\displaystyle \sf{\longrightarrow k\bigg[1-\bigg(\frac{11}{10} \bigg) \bigg]=-10}$

$\displaystyle \sf{\longrightarrow k=100}\:\:\:\:\: \cdots \sf{\color{orange}{ANSWER}}$

Therefore, the value of k is 100.

Answer 2

$\left(10\right)^9+2\left(11\right)^1\left(10\right)^8+3\left(11\right)^2\left(10\right)^7+10\left(11\right)^9=k\left(10\right)^9$

$\mathrm{Switch\:sides}$

$k\left(10\right)^9=\left(10\right)^9+2\left(11\right)^1\left(10\right)^8+3\left(11\right)^2\left(10\right)^7+10\left(11\right)$

$\mathrm{Apply\:rule}\:a^1=a$

$11^1=11$

$k\left(10\right)^9=10^9+10^8\cdot \:2\cdot \:11+10^7\cdot \:11^2\cdot \:3+11^9\cdot \:10$

$\mathrm{Multiply\:the\:numbers:}\:2\cdot \:11=22$

$k\left(10\right)^9=10^9+10^8\cdot \:22+10^7\cdot \:11^2\cdot \:3+11^9\cdot \:10$

$10^9=1000000000$

$22\cdot \:10^8=2200000000$

$3\cdot \:11^2\cdot \:10^7=3630000000$

$k\left(10\right)^9=1000000000+2200000000+3630000000+11^9\cdot \:10$

$\mathrm{Add\:the\:numbers:}\:1000000000+2200000000+3630000000=6830000000$

$k\left(10\right)^9=11^9\cdot \:10+6830000000$

$\mathrm{Divide\:both\:sides\:by\:}1000000000$

$\frac{k\cdot \:10^9}{1000000000}=\frac{11^9\cdot \:10}{1000000000}+\frac{6830000000}{1000000000}$

$\mathrm{Simplify}$

$k=\frac{11^9\cdot \:10+6830000000}{1000000000}$