Series Concept

A number series is formed by adding the terms of a number sequence in their given order. Two common examples are an arithmetic series, whose terms come from an arithmetic sequence, and a geometric series, whose terms come from a geometric sequence.

In the handshake example below, each new person adds another term to the sum.

Partial Sums Connect a Sequence to Its Series

For a sequence $(U_n)$, the sum of its first $n$ terms is the $n$th partial sum:

Finite problems ask for a particular partial sum. For an infinite series, we study whether the sequence $(S_n)$ approaches a finite limit as $n$ grows.

Number of Handshakes

Imagine there are several people in a group. If each person shakes hands exactly once with every other person in the group, how many total handshakes occur?

With three people, the first person shakes hands with the other two. One pair remains between the second and third people, so the total is $2+1=3$. For four people, the diagram contains one line for each pair. A crossing of lines does not add another person or handshake.

The table counts handshakes as one new person joins. Each newcomer shakes hands with everyone already present, so the numbers of additional handshakes are $1,2,3,4$.

Summations like $1+2+3+4$ are examples of a number series. In this case, the series is formed from the sum of the terms of the sequence of natural numbers $(1, 2, 3, 4, ...)$.

Each pair must be counted exactly once. Counting every person's handshakes independently would count each pair twice, once from each person's point of view.

A Second Counting Method

A handshake is an unordered pair of people. The number of pairs chosen from $n$ people is:

For $n=5$, this gives $\binom52=10$, matching the sequential sum.

Arithmetic and Geometric Patterns

The handshake totals $(1, 3, 6, 10, ...)$ are neither arithmetic nor geometric because consecutive terms have no constant difference or ratio ($3-1=2$, $6-3=3$, $10-6=4$).

Each expression $1$, $1+2$, $1+2+3$, and $1+2+3+4$ is a partial sum of the natural-number sequence. These partial sums show how a series is built from the terms of a sequence.

Practice

How many handshakes occur among $12 \text{ people}$ if every pair shakes hands exactly once?

The sequential-sum method gives:

The pair-counting formula gives the same result directly:

There are $66 \text{ handshakes}$.

Published: Apr 8, 2025. Updated: Sep 7, 2026.