Question

If (10)⁹ + 2(11)¹(10)⁸ + 3(11)²(10)⁷ +...+10(11)⁹ = k(10)⁹, then k is equals to:
( IIT JEE MAIN 2014 )
a. 121/10
b. 441/100
c. 100
d. 110

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Answer 1

Answer:

c. 100

Step-by-step explanation:

Given,

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

Dividing both sides by (10)^9

$1 + 2( \frac{11}{10} ) + 3 { (\frac{11}{10}) }^{2} + .... + 10 { (\frac{11}{10}) }^{9} = k \: \: ......(i)$

multiplying both sides by (11/10)

$( \frac{11}{10} ) + 2 {( \frac{11}{10} )}^{2} + 3 {( \frac{11}{10}) }^{3} + .... + 10 {( \frac{11}{10}) }^{10} = k( \frac{11}{10}) \: \: ....(ii)$

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

$(1 - \frac{11}{10} )k = 1 + \frac{11}{10} + {( \frac{11}{10}) }^{2} + {( \frac{11}{10} ) }^{3} + .... + { \frac{11}{10} )}^{9} - 10 {( \frac{11}{10} )}^{10}$

$= > (1 - \frac{11}{10} )k = \frac{1.( ({ \frac{11}{10}) }^{10} - 1)}{( \frac{11}{10} - 1)} - 10( { \frac{11}{10}) }^{10}$

$= > (1 - \frac{11}{10} )k = 10 { (\frac{11}{10} )}^{10} - 10 - 10 {( \frac{11}{10}) }^{10}$

$= > - \frac{1}{10}k = - 10$

$= > k = 100$

Hope this will help you