Question

find the sum of the series: 5/13 + 55/13^2 + 555/13^3 + 5555/13^4 + ..........upto infinity

Answer 1

HELLO DEAR,


Let S = 5/13 + 55/13² + 555/13³ + 5555/13⁴+ ------------( 1 )

Multiplying both sides by 1/13
we get,

1/13 S = 5/13² + 55/13³ + 555/13⁴ + ---------------( 2 )

Subtracting --------- ( 2 ) from ----------( 1 )

S - 1/13 S = (5/13 + 55/13² + 555/13³ + 5555/13⁴ + ............) - (5/13² + 55/13³ + 555/13⁴ + ........... )

we get,

12/13 S = 5/13 + (55/13² - 5/13²) + (555/13³ - 55/13³) + ( 5555/13⁴ - 555/13⁴) + ..............



12/13 S = 5/13 + 50/13² + 500/13³ + 5000/13⁴ + ..........

R.H.S is an infinite G.P. with first term a = 5/13 , common ratio r = 10/13

[As sum of infinite G.P. = a /(1 - r)]

12/13 S = (5/13) / (1 - 10/13)

12/13 S = (5/13)/(3/13)

12/13 S = 5/13 * 13/3

12/13 S = 5/3

S = 5/3 * 13/12

S = 65/36


I HOPE ITS HELP YOU DEAR,
THANKS.

Answer 2

65/36

I hope this helps you