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To calculate the formula for the sum of the first n terms of an AP, we go about it this way.
We know that the nth term of any AP can be written as a1+ (n-1) d.
Now, the sum of the first n terms S = a1+ a2 +a3 … an
=> S= a1+ a1 + d +a1+ 2d …a1 + (n-1) d
Or starting with the last term it can be written as
=> S= an + an-d + an-2d +…an- (n-1) d
Now adding the two forms
=> 2S = a1+ an + a1 + d + an-d +a1+ 2d + an-2d…a1 + (n-1) d + a1
all terms with d cancel
=> 2S = n (a1 + an)
=> S= (n/2) (a1+ an)
Now an = a1 + (n-1)*d
=> S = [2*a1 + (n-1)*d]*n/2
Therefore the sum of the first n terms is [2*a1 + (n-1)*d]*n/2 for all AP
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