Albert Teen YOU ARE LEARNING:  Arithmetic Sequences 1  # Arithmetic Sequences 1 ### Arithmetic Sequences 1

Arithmetic sequences are sequences where the difference between terms in the sequence are the same each time. Finding an unknown term in the sequence is called finding the nth term.

1

Is there a pattern in this sequence of numbers? Answer yes or no.  2

How big is the difference between $5$ and $8$?  3

How big is the difference between $8$ and $11$?  4

This is an example of an arithmetic sequence

There is a constant pattern in the sequence of numbers. The difference between each number and the following number is $+3$ 1

Is this an arithmetic sequence? Answer yes or no.  2

This is not an example of an arithmetic sequence

The numbers appear more or less random. There is no constant pattern in the sequence. 1

Is this an arithmetic sequence? Answer yes or no.  2

This is an example of an arithmetic sequence

There is a constant pattern. Each number is followed by a number that is $+12$ greater. What is the next term in this sequence? $4 \space\space\space 13 \space\space\space 22 \space\space\space 31 \space\space\space 40 \space\space\space ...$ 1

How many terms are listed in this sequence?  2

There are 5 terms in this sequence

You can label them term number $1$, $2$, $3$ , $4$ and $5$ and call that number label $n$ 3

What would be the 6th term in this sequence?  4

It's probably not hard to find the 7th, 8th and 9th terms either

But what if you wanted to find the 100th term?!

You can work out a formula that helps you find the value of any nth term in this sequence. 1

To work out this formula, you have to go through a few steps

First you work out how much bigger the value of the term gets every time $n$ gets 1 bigger. 2

What is the value of the term when $n=1$ in this sequence?  3

What is the value of the term when $n=2$?  4

How much bigger does the value of the term get every time $n$ gets $1$ bigger?  5

Every time $n$ gets one bigger the value of the term gets $+3$ bigger

So the first part of your formula is $3n$

But $3n$ is not the finished formula! You can call the missing piece of the formula $x$ 6

Now, what term value should the finished formula give you when $n=1$?  7

So you can substitute $1$ and $7$ into the formula like this $3 \times 1 + x = 7$. What is $x$?  8

So now you have the formula. Try testing it. What do you get if you let $n=5$?  9

What would be the value of the 50th term in this sequence?  1

What is the difference between the each of the numbers in this sequence?  2

So what do you multiply $n$ by in the formula for this sequence?  3

Now, let $n=1$ and work out what the $x$ in your formula should be.  4

To recap! You found the formula like this

You worked out the difference between each of the numbers in the sequence, which gave you $5n$

When $n=1$ the formula should give you $8$, so you worked out $x$ like this $5 \times 1 + x = 8$, which means $x=3$ 5

What is the 100th term in this sequence?  What is the 20th term in this sequence? $7 \space\space\space 11\space\space\space15\space\space\space19...$ 1

Summary! This is an arithmetic sequence

There is a constant pattern in the terms in the sequence. 2

Each term can be labelled by a number $n$

You use that $n$ to work out a formula for the sequence, so you can find the value of any nth term in the sequence. 3

To work out the formula, you first find the pattern between the terms

Here they get $+3$ bigger every time $n$ gets 1 bigger, so the first part of your formula is $3n$ 4

Then you find out what $x$ is by letting $n=1$

When $n=1$, the formula should give $5$, so you can work out $x$ like this $3 \times 1+x=5$.

That means that $x=2$ 5

You could now use this formula to work out any nth term

For example, the 10th term in this sequence would be $3 \times 10 +2 = 32$ 