1.

Find the sum of n terms of the sequence given by `a_(n)=(3^(n)+5n), n in N`.

Answer» Let the sum of n terms of the given sequence be `S_(n)`. Then,
`S_(n)=a_(1)+a_(2)+a_(3)+...+a_(n)`
`=(3^(1)+5xx1)+(3^(2)+5xx2)+(3^(3)+5xx2)+(3^(3)+5xx3)+...+(3^(n)+5xxn)`
`=(3+3^(2)+3^(3)+...+3^(n))+(5xx1+5xx2+5xx3+...+5xxn)`
`=(3+3^(2)+3^(3)+...+3^(n))+5xx(1+2+3+..+n)`
`=(3(3^(n)-1))/((3-1))+5xxn/2 (1+n)`
`=3/2 (3^(n)-1)+5/2 xx n (n+1)`.
Hence, the required sum is `3/2 (3^(n)-1)+5/2 n(n+1)`.


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