Question #315069

Represent the sum of first terms of the series 3+33+333+.........

using the sigma notation.



1
Expert's answer
2022-03-22T04:15:31-0400
S=3(1+11+111+1111+...nthterm).S= 3(1+11+111+1111+...n^{th} term).

Let the nth term be,

tn=1+10+100+...10n1tn=1+10+100+...10^{n-1}

This is a G.P. with first term 1 and common ratio of 10.

Thus, the sum of n terms,

S=3i=1ntn=3i=1n10n19S= 3\sum_{i=1}^{n} t_{n}=3\sum_{i=1}^{n}\frac{10^n-1}{9}

Hence,

S=310n9319S= 3\sum\frac{10^n}{9}-3\sum\frac{1}{9}

S=3[10(110n)(110)×9(19)×nS= 3[\frac{10(1-10^n)}{(1-10)\times9}-(\frac{1}{9})\times n

=3[10×10n181n9]= 3[10\times\frac{10^n-1}{81}-\frac{n}{9}]

Solving this we get,

S=[10×10n127n3]S= [10\times \frac{10^n-1}{27}-\frac{n}{3}]


Hence, the sum of first n terms of the series is,


=10n+1109n27= \frac{10^{n+1}-10-9n}{27}




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