Mathematics

Problem

Which one of the following gives the sum

S_n

of the first

n

terms of an arithmetic progression solely in terms of its first term

a

and

n

th term

a_n

? a)

n(a+a_n)

\dfrac{n}{2}(a_n-a)

\dfrac{n}{2}(a+a_n)

\dfrac{a+a_n}{2}

Intuitive explanation

The sum of an arithmetic progression depends on how many terms you're adding and their average value. Since the first and last terms are the extremes of the sequence, their average represents the typical value of all terms in between. Multiply this average by the count of terms to get the total sum—this formula works regardless of the common difference.

Intuitive explanation

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