Find the sum of $n$ terms of an AP whose $r^{t h}$ term is $(5 r + 1)$ .

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Solution

$r$ th term of the A.P. is $a_{r} =$ $(5 r + 1)$

Sum of $n$ terms $S_{n} = \sum_{r = 1}^{n} a_{r}$

$= \sum_{r = 1}^{n} 5 r + 1$

$\sum_{i = 1}^{n} k = 1 + 2 + 3 + \dots + n = \frac{n (n + 1)}{2}$ (sum of first n natural numbers)

$S_{n} = 5 \sum_{r = 1}^{n} r + \sum_{r = 1}^{n} 1$

$= 5 \times \frac{n (n + 1)}{2} + n$

$= \frac{5 n^{2} + 7 n}{2}$

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