S=3(1+11+111+1111+...nthterm).Let the nth term be,
tn=1+10+100+...10n−1
This is a G.P. with first term 1 and common ratio of 10.
Thus, the sum of n terms,
S=3∑i=1ntn=3∑i=1n910n−1
Hence,
S=3∑910n−3∑91
S=3[(1−10)×910(1−10n)−(91)×n
=3[10×8110n−1−9n]
Solving this we get,
S=[10×2710n−1−3n]
Hence, the sum of first n terms of the series is,
=2710n+1−10−9n
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