Question

For an AP if Sn- 3n2 - 4n, then: (i) the common difference of the AP is (A) 3 (B) 6 (ii) the first term of the AP is : (A) -1 (B) 2​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

Sum of n terms of an AP is

$\rm \: S_n = {3n}^{2} - 4n$

Let assume that

First term of an AP series is a

Common difference of an AP is d

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ Sum of n  terms of an arithmetic sequence is,

$\begin{gathered}\:\:{\underline{\orange{\boxed{\bf{\green{S_n\:=\dfrac{n}{2} \bigg(2 \:a\:+\:(n\:-\:1)\:d \bigg)}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

Sₙ is the sum of n terms of AP.

a is the first term of the sequence.

n is the no. of terms.

d is the common difference.

So, on substituting the values, we get

$\rm \: \dfrac{n}{2}\bigg(2a + (n - 1)d\bigg) = {3n}^{2} - 4n$

$\rm \: \dfrac{n}{2}\bigg(2a + (n - 1)d\bigg) = n(3n - 4)$

$\rm \: \dfrac{1}{2}\bigg(2a + (n - 1)d\bigg) = 3n - 4$

$\rm \: 2a + (n - 1)d= 6n - 8$

$\rm \: 2a + nd - d= 6n - 8$

can be re-arranged as

$\rm \: (2a - d) + nd= 6n - 8$

So, on comparing we get

$\rm\implies \:d \: = \: 6$

It means, option B is correct.

and

$\rm \: 2a - d = - 8$

$\rm \: 2a - 6 = - 8$

$\rm \: 2a = - 8 + 6$

$\rm \: 2a = - 2$

$\rm\implies \:a \: = \: - \: 1$

It means, option A is correct.

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Additional Information

↝ nᵗʰ term of an arithmetic sequence is,

$\begin{gathered}\:\:{\underline{\orange{\boxed{\bf{\green{a_n\:=\:a\:+\:(n\:-\:1)\:d}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

aₙ is the nᵗʰ term.

a is the first term of the sequence.

n is the no. of terms.

d is the common difference.