Common Difference of an AP from the Sum of n Terms

This short example finds the common difference $d$ of an arithmetic progression (AP) whose sum of the first $n$ terms is given by $S_n = n^2$.

Finding the first two partial sums

We are given

$S_n = n^2.$

Substituting $n = 1$ and $n = 2$ gives the sum of the first one and first two terms:

$S_1 = 1^2 = 1, \qquad S_2 = 2^2 = 4.$

Recovering the terms

The sum of one term is just the first term, so

$a_1 = S_1 = 1.$

Any term equals the difference of consecutive partial sums, $a_n = S_n - S_{n-1}$. For the second term:

$a_2 = S_2 - S_1 = 4 - 1 = 3.$

Finding the common difference

The common difference is the gap between consecutive terms:

$d = a_2 - a_1 = 3 - 1 = 2.$

So the common difference is $d = 2$.

Key takeaways

• When the sum $S_n$ is known, recover a term using $a_n = S_n - S_{n-1}$, and the first term from $a_1 = S_1$.
• The common difference is $d = a_2 - a_1$.
• For $S_n = n^2$, the first term is $1$ and the common difference is $2$.