Question

If 27 is the middle term of AP having 11 terms find the sum upto 11 terms

Answer 1

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

Given that

An AP series consists of 11 number of terms such that its middle term is 27.

We know, middle term of an AP if number of terms (n) are odd is given by (n + 1)/2 th term.

So, Here middle term is 6th term.

Let assume that

First term of an AP = a

Common difference = d

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,

$\rm :\longmapsto\:a_6 = a + (6 - 1)d$

$\rm :\longmapsto\:a + 5d = 27 - - - (1)$

Again,

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.

Thus,

Sum of 11 terms is given by

$\rm :\longmapsto\:S_{11} \: = \: \dfrac{11}{2}\bigg(2a + (11 - 1)d\bigg)$

$\rm :\longmapsto\:S_{11} \: = \: \dfrac{11}{2}\bigg(2a + (10)d\bigg)$

$\rm :\longmapsto\:S_{11} \: = \: 11 \times \bigg(a + 5d\bigg)$

$\rm :\longmapsto\:S_{11} \: = \: 11 \times 27$

$\bf :\longmapsto\:S_{11} \: = 297$