Question

The sum of the 'n'. terms of an arithmetic sequence is Sn =3n2+2n.
a) find its first term and common different.
b)write it's algebraic form

Answer 1

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

Given that,

Sum of n terms of an AP is

$\rm :\longmapsto\:S_n = {3n}^{2} + 2n$

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ Sum of n  terms of an arithmetic sequence is,

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

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.

Hence,

$\rm :\longmapsto\:S_n = {3n}^{2} + 2n$

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

$\rm :\longmapsto\:2a + (n - 1)d = 6n + 4$

$\rm :\longmapsto\:2a + nd - d = 6n + 4$

$\rm :\longmapsto\:(2a - d) + nd = 6n + 4$

On comparing we get,

$\red{\rm :\longmapsto\:d = 6}$

and

$\rm :\longmapsto\:2a - d = 4$

$\rm :\longmapsto\:2a - 6 = 4$

$\rm :\longmapsto\:2a = 4 + 6$

$\rm :\longmapsto\:2a = 10$

$\red{\bf :\longmapsto\:a = 5}$

We know,

↝ nᵗʰ term of an arithmetic sequence is,

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

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.

Tʜᴜs,

$\rm :\longmapsto\:a_n = 5 + (n - 1)6$

$\rm :\longmapsto\:a_n = 5 + 6n - 6$

$\rm :\longmapsto\:a_n = 6n - 1$

Hence,

$\rm :\longmapsto\:a_n = 6n - 1$

$\rm :\longmapsto\:a_1 = 5$

$\rm :\longmapsto\:d = 6$