Question

Find the sum to n terms: 8 + 88 + 888 + 8888 + …

Answer 1

S_n = 8 + 88 + 888 +  … upto n terms
= 8(1 + 11 + 111 + … upto n terms)

= `8/9` (9 + 99 + 999 + ... upto n terms)

= `8/9` [(10 – 1) + (100 – 1) + (1000 – 1) + ... upto terms)

= `8/9`[(10 + 100 +1000 + ... upto terms) – (1 + 1 + 1 ... n terms)]

But 10, 100, 1000, … n terms are in G.P. with

a = 10, r = `100/10` = 10

∴ S_n = `8/9[10((10^"n" - 1)/(10 - 1)) - "n"]`

= `8/9[10/9(10^"n" - 1) - "n"]`

∴ S_n = `8/81[10(10^"n" - 1) - 9"n"]`.