Find the sum of the following arithmetic progressions:

a + b, a − b, a − 3b, ... to 22 terms

#### Solution

In the given problem, we need to find the sum of terms for different arithmetic progressions. So, here we use the following formula for the sum of *n* terms of an A.P.,

`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

a + b, a − b, a − 3b, ... to 22 terms]

Common difference of the A.P. (d) = `a_2 - a_1`

= (a - b) -(a + b)

= a - b - a - b

= -2b

Number of terms* *(*n*) = 22

The first term for the given A.P. (*a*) = a + b

So, using the formula we get,

`S_22 = 22/2 [2(a + b) + (22- 1)(-2b)]`

`= (11)[2a + 2b + (21)(-2b)]`

`= (11)[2a + 2b - 42b]`

= 22a - 440b

Therefore the sum of first 22 terms for the give A.P is 22a - 440b