Question

If the sum of p terms of an A.P. is 4p2+ 3p, find its nth term.

Answer 1

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ Sum of n terms of an arithmetic sequence is,

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

Wʜᴇʀᴇ,

• Sₙ is the sum of n terms.

• a is the first term of the sequence.

• n is the no. of terms.

• d is the common difference.

Also,

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ 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.

Gɪᴠᴇɴ :

$\rm :\implies\:S_p \: = {4p}^{2} + 3p$

Tᴏ Fɪɴᴅ :

$\rm :\implies\:a_{n} \: term \: of \: the \: sequence.$

Cᴀʟᴄᴜʟᴀᴛɪᴏɴ :

$\green{\rm :\implies\:S_p \: = {4p}^{2} + 3p}$

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

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

$\rm :\implies\:2a + pd - d \: = 8p + 6$

$\rm :\implies\:(2a - d) + pd \: = 8p + 6$

On comparing, we get

$\rm :\implies\: \boxed{ \pink{ \bf \: d\: = \tt \: 8}}$

and

$\rm :\implies\:2a - d = 6$

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

$\rm :\implies\:2a = 14$

$\rm :\implies\: \boxed{ \pink{ \bf \: a \: = \tt \: 7 }}$

So,

↝ nᵗʰ term of an arithmetic sequence is,

$\rm :\implies\:a_{n} = a + (n - 1)d$

$\rm :\implies\:a_{n} = 7 + (n - 1)8$

$\rm :\implies\:a_{n} = 7 + 8n - 8$

$\rm :\implies\:a_{n} = 8n - 1$

Hence,  nᵗʰ term of an arithmetic sequence is,

$\rm :\implies\: \underline{ \large\boxed{ \pink{ \bf \: a_{n}\: = \tt \: 8n - 1}}}$