Question

if the sum of an AP 3n^2 +4n then the 10th term is ________

Answer 1

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

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ Sum of first 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 first n terms.

• a is the first term of the sequence.

• n is the no. of terms.

• d is the common difference.

Cᴀʟᴄᴜʟᴀᴛɪᴏɴ :-

Given that

$\red{\rm :\longmapsto\:S_n = {3n}^{2} + 4n}$

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

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

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

On comparing, we get

$\rm :\implies\:\purple{\boxed{ \bf \:d \: = \: 6}}$

And

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

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

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

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

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

Now,

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.

Tʜᴜs,

$\rm :\longmapsto\:a_{10} = a + (10 - 1)d$

$\rm :\longmapsto\:a_{10} = a + 9d$

$\rm :\longmapsto\:a_{10} = 7 + 9 \times 6$

$\rm :\longmapsto\:a_{10} = 7 + 54$

$\rm :\longmapsto\:a_{10} =61$