Find the sum of the series:

Question:

Find the sum of the series:

3 + 5 + 7 + 6 + 9 + 12 + 9 + 13 + 17 + ... to 3n terms.

Solution:

The given sequence i.e., 3 + 5 + 7 + 6 + 9 + 12 + 9 + 13 + 17 +..... to 3n terms.

can be rewritten as 3 + 6 + 9 + .... to n terms + 5 + 9 + 13 + .... to n terms + 7 + 12 + 17 + .... to n terms

Clearly, all these sequence forms an A.P. having n terms with first terms 3, 5, 7 and common difference 3, 4, 5

Hence, required sum $=\frac{n}{2}[2 \times 3+(n-1) 3]+\frac{n}{2}[2 \times 5+(n-1) 4]+\frac{n}{2}[2 \times 7+(n-1) 5]$

$=\frac{n}{2}[(6+3 n-3)+(10+4 n-4)+(14+5 n-5)]$

$=\frac{n}{2}[12 n+18]$

$=3 n(2 n+3)$

 

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