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Arithmetic sequence

Calculus
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Series and Sequences
schedule Jul 1, 2022
Last updated
local_offer Arthur
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In an arithmetic sequence, each term is the previous term summed with a constant. The constant we add to each term to get the next term is referred to as the common difference.

General form

The general form of an arithmetic sequence can be expressed as follows:

$$\{a, a + d, a+2d, ..., a+(n-1)d\}$$
$$1st\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,nth \,\, \text{term}$$

where:

  • $a$ is the first term

  • $d$ is the common difference between the terms

Example

Consider the following arithmetic sequence:

$$\{2, 6,10,14,18,22,...\}$$

Here we start with 2 and add 4 to get the next term in the arithmetic sequence.

Therefore in terms of the general form of the arithmetic sequence we can say:

$a$ = 2 (first term)

$d$ = 4 (common difference)

Sum of arithmetic series

To compute the sum $S_n$ of an arithmetic series:

$$S_n = a + (a+d) + \cdots + (a+(n-1)d)$$

We can use the following formula:

$$S_n = \frac{n[a+a_n]}{2}$$

Example

Question. To calculate $S_n$:

$$\{2,6,10,14,18,22\}$$

Solution. Comparing to the general form we can see that $a=2$, $d=4$ and $n=6$. Now plugging these numbers into the formula for sum of arithmetic series:

$$\begin{align*} S_n &= \frac{6[2 + 22]}{2} \\ &= 72 \end{align*}$$

Derivation

We can write the equation for sum of an arithmetic series in the following two ways:

$$\begin{align*} S_n &= {\color{red}a} +{\color{blue}(a+d)}+ \cdots + {\color{green}(a+(n-1)d)} \\ S_n &={\color{red}[a+(n-1)d]} +{\color{blue}[a+(n-2)d]}+\cdots+{\color{green}a} \end{align*}$$

Adding the two equations together:

$$2S_n=[2a+nd-d]+[2a+nd-d]+⋯+[2a+nd-d]$$

Note that there are $n$ terms altogether so the above simplifies to:

$$\begin{align*} 2S_n &= n[2a+nd-d] \\ S_n &= \frac{n[2a+nd-d]}{2} \\ S_n &= \frac{n[a+ {\color{red}a + (n-1)d}]}{2} \end{align*}$$

The key is that the red component can be re-written as:

$$S_n = \frac{n[a+ {\color{red}a_n}]}{2}$$

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Published by Arthur Yanagisawa
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