Question

Solve previous year Question of iit jee

Chapter :- sequence and series​

Answer 1

Answer:

let S = 0.7 + 0.77 + 0.777 + . . . . . upto 20 term.

= 7(0.1 + 0.11 + 0.111 + . . . . . . .upto 20 term)

multiply and divide by 9 :

$= \frac{7}{9} (0.9 + 0.99 + 0.999 + . \: . \: . \: . \: upto \: 20 \: term)$

$= \frac{7}{9} ((1 - 0.1) + (1 - 0.01) + (1 - 0.001) + . \: . \: . \: .upto \: 20 \: term)$

taking 1 common in each . ..

$= \frac{7}{9} (20 \times 1- (0 .1 + 0.01+ 0.001 + ... \: upto \: 20 \: term))$

now, 0.1 , 0.01 ,0.001 are in gp with common ratio = 0.1

$= \frac{7}{9} (20 - ( \frac{0.1(1 - {0.1}^{20}) }{1 - 0.1} ))$

$= \frac{7}{9} (20 - \frac{0.1(1 - \frac{1}{ {10}^{20} } ) }{0.9} )$

$= \frac{7}{9} (20 - \frac{1 - {10}^{ - 20} }{9} )$

$= \frac{7}{9} ( \frac{180 - (1 - {10}^{ - 20} )}{9} )$

$= \frac{7(180 - 1 + {10}^{ - 20}) }{81}$

$= \frac{7(179 + {10}^{ - 20}) }{81}$

hence, correct option is (c).