Question

if the sum of first p terms of an ap is 3n square + 5n find it common difference

Answer 1

$\underline{\bold{Solution:-}}$

Let d be the common difference.

Note:- Here u have by mistake type p instead of n.

$Sum \: of \: first \: {n}^{th} \: term = 3 {n}^{2} + 5n$

For finding the sum of 1 term , use n = 1.

$S_{1} = a_{1} = 3 \times {1}^{2} + 5 \times 1 \\ \\ = 3 + 5 \\ \\ = 8$

For finding the sum of 2 terms , use n = 2

$S_{2} = 3 \times {2}^{2} + 5 \times 2 \\ \\ = 3 \times 4 + 10 \\ \\ = 12 + 10 \\ \\ = 22$

We know that,

$a_{1} + a_{2} = S_{2} \\ \\ 8 + a_{2} = 22 \\ \\ a_{2} = 22 - 8 \\ \\ a_{2} = 14 \\ \\ 8 + (2 - 1)d = 14 \\ \\ d = 14 - 8 \\ \\ \boxed{d = 6}$

So, the common difference is 6.

Answer 2

Note: Your questions seems to be incorrect.

For p terms, it should be 3p^2 + 5p

              (or)

For n terms, it should be 3n^2 + 5n.

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I am considering for n terms.

Given that Sum of first n terms of an AP is 3n^2 + 5n.

When n = 1:

⇒ S₁ = a₁ = 3(1)^2 + 5(1)

               = 8.

So, first term = a₁ = 8.

When n = 2:

⇒ S₂ = 3(2)^2 + 5(2)

        = 12 + 10

        = 22.

So,Second term a₂ = S₂ - S₁

                                = 22 - 8

                                = 14.

Common difference d = a₂ - a₁

                                     = 14 - 8

                                     = 6.

Therefore, common difference d = 6.

Hope it helps!