Question

Find the sum of the first $22$ terms of an AP in which $d=7$ and the $22nd$ term is $149$.

Answer 1

Step 1: Find the first term of the series

Given that, common difference, $d=7$

${22}^{nd}$ term, $a_{22}=149$

Sum of the first $22$term, $S_{22}=?$

Using the formula of$n$th term, we get

$$\begin{gathered} a_n=a+(n-1)d \\ \Rightarrow a_{22}=a+(22-1)d \\ \Rightarrow 149=a+21\times 7 \\ \Rightarrow 149=a+147 \\ \Rightarrow a=2 \end{gathered}$$

Step 2: Find the sum of the terms

Sum of $n$ terms,

$$\begin{gathered} \Rightarrow S_n=\frac{n}{2}(a+a_n) \\ \Rightarrow S_{22}=\frac{22}{2}(2+149) \\ \Rightarrow S_{22}=11\times 151 \\ \Rightarrow S_{22}=1661 \end{gathered}$$

Hence, the Sum of first $22$ terms is $1661$.