Question:
Very-Short and Short-Answer Questions
If the sum of first n terms is (3n2 + 5n), find its common difference.
Solution:
Let Sn denotes the sum of first n terms of the AP.
$\therefore S_{n}=3 n^{2}+5 n$
$\Rightarrow S_{n-1}=3(n-1)^{2}+5(n-1)$
$=3\left(n^{2}-2 n+1\right)+5(n-1)$
$=3 n^{2}-n-2$
Now,
$n^{\text {th }}$ term of the $\mathrm{AP}, a_{n}=S_{n}-S_{n-1}$
$=\left(3 n^{2}+5 n\right)-\left(3 n^{2}-n-2\right)$
$=6 n+2$
Let d be the common difference of the AP.
$\therefore d=a_{n}-a_{n-1}$
$=(6 n+2)-[6(n-1)+2]$
$=6 n+2-6(n-1)-2$
$=6$
Hence, the common difference of the AP is 6.
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