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Question
Find the sum of the following arithmetic progressions:
a + b, a − b, a − 3b, ... to 22 terms
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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
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