Difference between Arithmetic and Geometric Series

Arithmetic Series

Basic concept:

An arithmetic series is the sum of the terms of an arithmetic sequence. Remember, an arithmetic sequence is one that has a constant difference (common difference) between its terms ($b$).

So, we are summing terms with the pattern: $a, a+b, a+2b, a+3b, \dots$.

The sum of the first $n$ terms ($S_n$) of an arithmetic series can be calculated using the formula:

or

Where $a$ is the first term and $U_n$ is the $n$-th term.

Simple analogy:

Imagine you are stacking bricks. The first layer has 1 brick, the second layer has 3 bricks, the third layer has 5 bricks, and so on (common difference = 2). An arithmetic series represents the total number of bricks needed to make a stack $n$ layers high.

Geometric Series

Basic concept:

A geometric series is the sum of the terms of a geometric sequence. Remember, a geometric sequence is one that has a constant ratio (common ratio) between its terms ($r$).

So, we are summing terms with the pattern: $a, ar, ar^2, ar^3, \dots$.

The sum of the first $n$ terms ($S_n$) of a geometric series can be calculated using the formula:

for $r \neq 1$, where $a$ is the first term and $r$ is the ratio.

Simple analogy:

Going back to the example of bacteria dividing (1 becomes 2, 2 becomes 4, etc., ratio = 2). A geometric series is the total number of bacteria after $n$ divisions. For example, the total number of bacteria after 3 divisions is $1 + 2 + 4 = 7$.

Key Differences