1Cademy - Sum of the First n Terms of an Arithmetic Sequence

Learn Before

Arithmetic Sequence

Formula

Sum of the First n Terms of an Arithmetic Sequence

The sum of the first $n$ terms of an arithmetic sequence, denoted $S_n$ , is the total obtained by adding all terms from $a_1$ through $a_n$ : $S_n = a_1 + a_2 + a_3 + \dots + a_n$ . Rather than adding each term individually, a closed-form formula allows $S_n$ to be computed directly using only the first term, the last term, and the number of terms:

$S_n = \frac{n}{2}(a_1 + a_n)$

where $a_1$ is the first term and $a_n$ is the $n$ th (last) term being summed.

This formula is derived by writing the sum twice — once in forward order starting from $a_1$ and once in reverse order starting from $a_n$ — and then adding the two expressions together. In the forward direction, each successive term increases by the common difference $d$ : $S_n = a_1 + (a_1 + d) + (a_1 + 2d) + \dots + a_n$ . In the reverse direction, each successive term decreases by $d$ : $S_n = a_n + (a_n - d) + (a_n - 2d) + \dots + a_1$ . When these two expressions are added term by term, the $d$ terms cancel and every pair sums to $(a_1 + a_n)$ . Since there are $n$ such pairs, the result is $2S_n = n(a_1 + a_n)$ . Dividing both sides by $2$ yields the formula $S_n = \frac{n}{2}(a_1 + a_n)$ .

0

1

Updated 2026-05-25

Contributors are:

Who are from:

References

Learn Before

Related

Learn After