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Question
Find 2 + 22 + 222 + 2222 + … upto n terms.
Solution
Sn = 2 + 22 + 222 +… upto n terms
= 2(1 + 11 + 111+ … upto n terms)
= `2/9` (9 + 99 + 999 + upto n terms)
= `2/9`[(10 – 1) + (100 – 1) + (1000 – 1) + ... upto n terms]
= `2/9`[(10 – + 100 + 100 + ... upto n terms) – (1 + 1 + 1 ... n terms)]
Since, 10, 100,1000, … n terms are in G.P. with a= 10, r = `100/100` = 10
∴ Sn = `2/9[10((10^"n" - 1)/(10 - 1)) - "n"]`
= `2/9[10/9(10^"n" - 1) - "n"]`
∴ Sn = `2/81[10(10^"n" - 1) - 9"n"]`
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