Sequence and Series

You will often encounter sequence and series in evaluating Big-O equations.

Geometric series

In geometric series, there are common ratio ($r$) between terms.

3, 6, 12, 24, 48...

$r=\frac{a_2}{a_1}=2$

Geometric series formula

When the number of terms is known, the sum of the terms $S$ can be described as: $S_n=\frac{a(r^n-1)}{r-1}$ Where:

• $n$ is the number of terms
• $r$ is the common ratio
• $a$ is the first term

When the number of terms is unknown, the sum of the terms can be described as: $S=\frac{a}{1-r}$

Arithmetic series

In arithmetic series, there are common difference ($d$) between terms.

5, 8, 11, 14, 17...

$d=a_2-a_1=3$

When the number of terms is known, the sum of the terms $S$ can be described as: $S_n=\frac{n}{2}\times (a+l)$ Where:

• $n$ is the number of terms
• $a$ is the first term
• $l$ is the last term

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