For a finite given by where a1 is the first term, d is the , and n is the number of terms, the sum of all terms, can be calculated using the following formula.
To explain why the sum is determined this way, consider the arithmetic sum
Instead of adding all terms in one go, note that the first and last term can be rewritten as 2
fives. The second and second to last term can be rewritten in the same way. In fact, all opposite pairs can be rewritten as 2
Now, there are 9
fives, and the sum is 9⋅5.
is the of the first and last term, which means that it can be written as
This is multiplied by 9,
which is the number of terms, n,
leading to the formula