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Arithmetic Progression — Maths STD 10 — Question

Gujarat BoardEnglish MediumSTD 10MathsArithmetic Progression4 Marks
Question
If the sum of first n term is $(3n^2+ 5n),$ find its common difference.
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Answer

Let $S_n$ 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,
$\mathrm{n}^{\text {th }} \text { term of the } A P, a_n=S_n-S_{n-1}$
$ =\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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