Arithmetic & Geometric Series

A sequence is a list of numbers.
A series is the sum of the terms of a sequence.

For example:

Sequence:

$$2 , 5 , 8 , 11 , \ldots$$

Series:

$$2 + 5 + 8 + 11 + \ldots$$

We study two important types:

• Arithmetic series

• Geometric series

These occur in algebra, finance, physics, and especially in Calculus BC infinite series.

Arithmetic Series​

Formula for Arithmetic Series

There are two forms, both equal:

$$S_{n}=\frac{n}{2}(a_{1}+a_{n})$$​

$$S_{n}=\frac{n}{2}[2a_{1}+(n-1)d]$$

Example

Find the sum of the first 20 terms of:

$$3, 7, 11, 15 , \ldots$$

then:

$$first term a_{1}=3 , common difference d = 7 – 3 =4$$

$$S_{20}=\frac{20}{2}[2(3)+(20-1)]$$

$$S_{20}=10(6+76)=10(82)=820$$

Geometric Series

Formula for Finite Geometric Series

$$S_{n}=a_{1}\frac{1-r^{n}}{1-r}$$​

or equivalently:

$$S_{n}=a_{1}\frac{r^{n}-1}{r-1}$$​​​

Example

Find the sum of the first 6 terms:

$$5 , 10 , 20 , 40 , \ldots$$

$$first term a_{1} = 5 , common ratio r = 2$$

$$S_{6}=5\cdot\frac{1-2^{6}}{1-2}=5\frac{1-64}{-1}=5(63)=315$$

Infinite Geometric Series

(Very important for AP Calculus BC)

An infinite geometric series:

$$a_{1}+a_{1}r+a_{1}r^{2}+a_{1}r^{3}+\ldots$$

Converges only if:

$$|r|<1$$

If it converges, the sum is:

$$S=\frac{a_{1}}{1-r}$$​​​

Example

$$1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots$$

Here:

$$first term a_{1}=1 , common ratio r = \frac{1}{2}$$

$$S=\frac{1}{1-\frac{1}{2}}=\frac{1}{\frac{1}{2}}=2$$

This idea leads into power series and Taylor series in Calculus BC.