Question

Find the sum of the series:
3 + 5 + 7 + 6 + 9 + 12 + 9 + 13 + 17 + ... to 3n terms.

Answer 1

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 = $\frac{n}{2}\left[2\times 3+\left(n-1\right)3\right]+\frac{n}{2}\left[2\times 5+\left(n-1\right)4\right]+\frac{n}{2}\left[2\times 7+\left(n-1\right)5\right]$

$$\begin{gathered} =\frac{n}{2}\left[\left(6+3n-3\right)+\left(10+4n-4\right)+\left(14+5n-5\right)\right] \\ =\frac{n}{2}\left[12n+18\right] \\ =3n\left(2n+3\right) \end{gathered}$$