Question

If `(10)^9+""2(11)^1(10)^8+""3(11)^2(10)^7+""ddot""+""10(11)^9=k(10)^9` , then k is equal to (1) `(121)/(10)` (2) `(441)/(100)` (3) 100 (4) 110

Answer 1

`(10)^9[1 + 2(11/10)^1 + 3 (11/10)^2 + ....+ (11/10)^9]`
`k = 1 + 2*11/10 + 3*(11/10)^2 + ... + 100(11/10)^9`
also,`11/10k = 11/10 + 2(11/10)^2 + .......... + 9(11/10)^9 + 10 (11/10)^2 `
subtracting eqn`(1)-(2):`
`-k/10 = 1 + 11/10 + (11/10)^2 + ...... + (11/10))^9 - 10(11/10)^10`
`S_n = (a(1-r^n))/(1-r)`
`= 1 xx((11/10)^10 -1)/(11/10 - 1)`
`-k/10 = 10 [(11/10)^10 - 1] - 10(11/10)^10`
`= 10 xx(11/10)^10 - 10 -10(11/10)^10`
`-k/10 = -10`
`k=100`
option 3 is correct