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Arithmetic sequences

Arithmetic sequences

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Explainer Video

Tutor: Labib

Summary

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 nthn^{th}nth term, unu_nun​​, of an arithmetic sequence can be given by an equation where aaa is the first term and ddd is the common difference. 

​un=a+(n−1)d\boxed {u_n=a+(n-1)d}un​=a+(n−1)d​​​


Note: The common difference can be positive or negative. 


Example 1

An arithmetic sequence is given by the formula

​un=5−8(n−1)u_n=5-8(n-1)un​=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 5‾\underline55​ and that the common difference is −8‾\underline{-8}−8​.


Thus the first four terms are:

5,−3,−11,−19‾\underline{5,-3,-11,-19}5,−3,−11,−19​​​


Example 2

The fifth term of an arithmetic sequence is 888 and the ninth term is 232323. What is the first term, aaa, and the common difference, ddd? Find the 101th101^{th}101th term.


Express the fifth and ninth term algebraically:

​5th term:u5=a+(5−1)d=a+4d=89th term:u9=a+(9−1)d=a+8d=23\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}5th term:u5​9th term:u9​​=a+(5−1)d=a+4d=8=a+(9−1)d=a+8d=23​​​


Solve the equations simultaneously: 

a+8d=23a+4d=84d=15\begin{aligned} a+8d &= 23 \\ a+4d &= 8 \\ \hline 4d &= 15 \\ \end{aligned} a+8da+4d4d​=23=8=15​​​​

d=3.75‾\underline{d= 3.75} d=3.75​​​


Solve for aaa:

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

a=7‾\underline {a = 7}a=7​​​


Find the 101th101^{th}101th term:

u101=a+(n−1)d=7+(n−1)(3.75)u_{101}=a+(n-1)d = 7+ (n-1)(3.75)u101​=a+(n−1)d=7+(n−1)(3.75)​​

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

101th term: u101=382‾\underline{101^{th} \textit{ term: } u_{101} = 382}101th term: u101​=382​​​



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FAQs - Frequently Asked Questions

What is the notation for the first term in a sequence?

The first term of a sequence is labelled a.

What is an arithmetic sequence?

A sequence where each term has the same common difference between each other.

How do you find a given term of an arithmetic sequence?

Use the formula a+(n-1)d.

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