What is the sum of the first n terms of the AP a, 3a, 5a,..

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Solution

The given AP is a, 3a, 5a,..

Here,

First term, a = a

Common difference, D = 3a-a = 2a

Therefore, $S_{n}$ i.e sum of first n terms

$S_{n} = \frac{n}{2} [2 a + (n - 1) d]$

$= \frac{n}{2} [2 \times a + (n - 1) \times 2 a]$

$= \frac{n}{2} (2 a + 2 a n - 2 a) = \frac{n}{2} \times 2 a n = a n^{2}$

Hence, the required sum is $a n^{2}$

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What is the sum of the first n terms of the AP a, 3a, 5a,..

The given AP is a, 3a, 5a,.. Here, First term, a = a Common difference, D = 3a-a = 2a Therefore,Sn i.e sum of first n terms Sn=n2[2a+(n−1)d] =n2[2×a+(n−1)×2a] =n2(2a+2an−2a)=n2×2an=an2 Hence, the required sum is an2