Sums of numerical sequences and difference series

Given a sequence $(a_n)$, the sum of the first $n$ terms is $S_n = a_1 + a_2 + ext{...} + a_n$. The difference series (or first differences) is the sequence $(d_n)$ where $d_n = a_{n+1} - a_n$. Recognizing patterns in the difference series can help us find formulas for the original sequence or sum.

Important properties

• If the difference series is constant, the original sequence is arithmetic.

• The sum of an arithmetic sequence can be found using $S_n = \frac{n}{2}(a_1 + a_n)$.

• If the difference series itself has a pattern (like being arithmetic), the original sequence may be quadratic.

• Summing difference series can help telescope complicated sums.