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, `a_22 = 149` and difference d = 22

we know `a_n = a + (n - 1)d`

22 nd term, `a_22 = a + (22 - 1)d`

`=> 149 = a + 21 xx 22`

`=> a = 149 - 462`

`=> a = - 313`

We know, sum of n terms

`S_n = n/2 [2a + (n - 1)d]`

`=> S_22 = 22/2[2(-313) + (22 - 1)22]`

`=> S_22 = 11[-626 + 21 xx 22]`

`=> S_22 = 11[-626 + 462]`

`=> S_22 = 11 xx -164`

`=> 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

`S_n = 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

\[S_{22} = \frac{22}{2}\left\{ 2 \times \left( - 313) + (22 - 1) \times 22 \right) \right\}\]
\[ S_{22} = 11\left\{ - 626 + 462 \right\}\]
\[ S_{22} = - 1804\]

Hence, the sum of 22 terms is −1804.