#### Questions

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.

#### Solution 1

Given d = 22,

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.

#### Solution 2

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