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Question
Find the sum to n terms: 8 + 88 + 888 + 8888 + …
Solution
Sn = 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
∴ Sn = `8/9[10((10^"n" - 1)/(10 - 1)) - "n"]`
= `8/9[10/9(10^"n" - 1) - "n"]`
∴ Sn = `8/81[10(10^"n" - 1) - 9"n"]`.
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