Arithmetic sequences

In a nutshell

An arithmetic sequence is one where there is a common difference between each term.

Using arithmetic sequences

The common difference of an arithmetic sequence can be used to find any term in the sequence. The $n^{th}$ term, $u_n$ , of an arithmetic sequence can be given by an equation where $a$ is the first term and $d$ is the common difference.

$\boxed {u_n=a+(n-1)d}$

Note: The common difference can be positive or negative.

Example 1

An arithmetic sequence is given by the formula

$u_n=5-8(n-1)$

What is the first term of the sequence and what is the common difference between each term? Hence give the first four terms in the sequence.

By comparing to the general formula above, you have that the first term is $\underline5$ and that the common difference is $\underline{-8}$ .

Thus the first four terms are:

$\underline{5,-3,-11,-19}$ 

Example 2

The fifth term of an arithmetic sequence is $8$ and the ninth term is $23$ . What is the first term, $a$ , and the common difference, $d$ ? Find the $101^{th}$ term.

Express the fifth and ninth term algebraically:

$\begin{aligned} 5^{th} \ \textit{term}: u_5 &= a + (5-1)d = a + 4d = 8 \\ 9^{th} \ \textit{term}: u_9 &= a + (9-1)d = a+8d =23 \end{aligned}$

Solve the equations simultaneously:

$\begin{aligned} a+8d &= 23 \\ a+4d &= 8 \\ \hline 4d &= 15 \\ \end{aligned}$ 

$\underline{d= 3.75}$ 

Solve for $a$ :

$\begin{aligned} a + 4(3.75) &= 8 \\ a+ 15 &= 8 \end{aligned}$ 

$\underline {a = 7}$ 

Find the $101^{th}$ term:

$u_{101}=a+(n-1)d = 7+ (n-1)(3.75)$ 

$\space=7 + (101-1)(3.75) = 7 + 375$ 

$\underline{101^{th} \textit{ term: } u_{101} = 382}$ 