Question

Find the sum of first 22 terms of an A.P. in which d = 22 and a = 149.

Let there be an A.P. with first term 'a', common difference 'd'. If a_n denotes in n^th term and S_n the sum of first n terms, find.

$$S_{22},\text{ if }d=22\text{ and }{a}_{22}=149$$

 

Answer 1

Given 22nd term, ${\displaystyle a_{22}=149}$ and difference d = 22

we know ${\displaystyle a_n=a+(n-1)d}$

22 nd term, ${\displaystyle a_{22}=a+(22-1)d}$

${\displaystyle \Rightarrow 149=a+21\times 22}$

${\displaystyle \Rightarrow a=149-462}$

${\displaystyle \Rightarrow a=-313}$

We know, sum of n terms

${\displaystyle S_n=\frac{n}{2}[2a+(n-1)d]}$

${\displaystyle \Rightarrow S_{22}=\frac{22}{2}[2(-313)+(22-1)22]}$

${\displaystyle \Rightarrow S_{22}=11[-626+21\times 22]}$

${\displaystyle \Rightarrow S_{22}=11[-626+462]}$

${\displaystyle \Rightarrow S_{22}=11\times -164}$

${\displaystyle \Rightarrow S_{22}=-1804}$

Hence sum of 22 terms -1804

Answer 2

Given d = 22, 

$$a_{22}=149$$

We know that

a_n = a + (n-1)d

$$149=a+(22-1)22$$
$$149=a+462$$
$$a=-313$$

Now, Sum is given by

${\displaystyle S_n=\frac{n}{2}[2a+(n-1)d]}$

Where; a = first term for the given A.P.

d = common difference of the given A.P.

n = number of terms

So, using the formula for n = 22, we get

$$S_{22}=\frac{22}{2}\{2\times (-313)+(22-1)\times 22)\}$$
$$S_{22}=11\{-626+462\}$$
$$S_{22}=-1804$$

Hence, the sum of 22 terms is −1804.