Question

10) The 7 th term of an A.P is 32. a Find the sum of the first term and 13 th term?. b Find the sum of the first 13 terms?​

Answer 1

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

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,

↝ 7ᵗʰ term is 32

$\rm :\longmapsto\:a + (7 - 1)d = 32$

$\rm :\longmapsto\:\boxed{ \tt{ \: a + 6d = 32}} \: - - - (1)$

Hence,

$\rm :\longmapsto\:a_1 + a_{13}$

$\rm \:  =  \:a + a + (13 - 1)d$

$\rm \:  =  \:2a + 12d$

$\rm \:  =  \:2(a + 6d)$

$\rm \:  =  \:2 \times 32$

$\rm \:  =  \:64$

Thus,

$\rm :\longmapsto\:\boxed{ \tt{ \: a_1 + a_{13} = 64 \: \: }}$

Now,

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.

So,

$\rm \longmapsto\:S_{13}\:=\dfrac{13}{2} \bigg(2 \:a\:+\:(13\:-\:1)\:d \bigg)$

$\rm \longmapsto\:S_{13}\:=\dfrac{13}{2} \bigg(2 \:a\:+\:(12)\:d \bigg)$

$\rm \longmapsto\:S_{13}\:=\dfrac{13}{2} \times 2\bigg( \:a\:+\:6\:d \bigg)$

$\rm \longmapsto\:S_{13}\:=13 \times 32$

$\rm \longmapsto\:S_{13}\:= \: 416$

Hence,

$\rm \longmapsto\:\boxed{ \tt{ \: \: S_{13}\:= \: 416 \: \: }}$