Answer :

S = n [First term is 1, common ratio 1; so sum to n terms = 1 + 1 + 1 + - - = n]
ii) S
= (2ⁿ - 1)/(2 - 1) = (2ⁿ - 1)
iii) S
= (3ⁿ - 1)/2
iv) S
= (4ⁿ - 1)/3 ..
v) So, S
+ S + 2S + 3S + - - - - + (n - 1)Sⁿ =
= n + (2ⁿ - 1) + (3ⁿ - 1) + (4ⁿ - 1) + - - - - - - - + (nⁿ - 1)
= n + ( - 1 - 1 - 1 .... to n - 1 terms) + (2ⁿ + 3ⁿ + 4ⁿ + .... + nⁿ) =
= n - (n - 1) + (2ⁿ + 3ⁿ + 4ⁿ + .... + nⁿ) = 1 + (2ⁿ + 3ⁿ + 4ⁿ + .... + nⁿ)
= 1ⁿ + 2ⁿ + 3ⁿ + 4ⁿ + .... + nⁿ [Proved]


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