**Arithmetic Sequence** *KS3 Revision*

## What you need to know

**Things to remember:**

- Sequences often involve substitution.
- Both numbers in a sequence can be negative.
- $n$ is the term position in the sequence.

Before we start, we need to know two important pieces of terminology:

**Sequences**just refers to the order that things appear in.**Arithmetic sequences**are sequences of numbers that go up or down by the same amount each time.

Sequences can come in all shapes and sizes:

**The alphabet**

a b c d e f g

We know that the next letters will be h, I, j, l, etc.

**Sides of shapes**

We can see that each time there is one more side on the shape, so the next ones will be heptagon, octagon, nonagon, etc.

**Numbers**

1 2 3 4 5 6 7 8 9 10

We can see that this is just the 1 times table, so the next numbers will be 11, 12, 13, etc.

50 45 40 35 30 25 20

We can see that this is just the 5 times table going down, so the next numbers will be 15, 10, 5, etc.

The last two sequences are a special kind, which we refer to as “arithmetic sequences”. Arithmetic sequences go up or down by the same amount each time.

*Find the next three terms in the sequence following sequence:*

*2 9 16 23 30 37*

* *

We will do this in two steps:

**Step 1: **Find out how much it changes by each time by choosing two numbers next to each other and subtracting the one on the left from the one on the right.

*Hint: **Try doing it with more than one pair to make sure you have done it correctly.*

16-9=7

30-23=7

So, it changes by 7 each time.

**Step 2: **Add this difference to the last term to find the next terms.

30+7=37

37+7=44

44+7=51

So, the next three terms are 37, 44, and 51

*Find the next three terms in the sequence following sequence:*

*25 22 19 16 13 10 7*

** ****Step 1: **Find out how much it changes by each time by choosing two numbers next to each other and subtracting the one on the left from the one on the right.

*Hint: **Try doing it with more than one pair to make sure you have done it correctly.*

22-25=-3

10-13=-3

So, it changes by -3 each time.

**Step 2: **Add this difference to the last term to find the next terms.

*Hint: **When we add a negative, it is the same as just subtracting.*

7-3=4

4-3=1

1-3=-2

So, the next three terms are 4, 1, and -2.

We can also write arithmetic sequences as algebraic expressions. Let’s see how we do this and why.

*If n is the place in the sequences, find the first 4 terms in the sequence given by 4n-1.*

First term – n=1

Second term – n=2

Third term – n=3

Third term – n=

So, to find our first three terms, we just need to substitute in these values of n

First term – n=1\rightarrow4\times1-1=4-1=3

Second term – n=2\rightarrow4\times2-1=8-1=7

Third term – n=3\rightarrow4\times3-1=12-1=11

Fourth term – n=4\rightarrow4\times4-1=16-1=15

So, the first four terms in the sequence are 3, 7, 11 and 15. If we look closely, we can see that this sequence is going up by 4 each time, which is the same number before the variable in the algebraic expression! Notice how if we subtract this from the first term, 3-4=-1 we find the number by itself in the algebraic expression.

*Find the algebraic expression for the following sequence:*

*7 16 25 34 43 52*

We will find the algebraic expression in three steps, using the points we outline above.

**Step 1: **Find how much the sequences changes by each time, this is usually referred to as the common difference.

52-43=9

34-24=9

The common difference here is 9, and this goes in front of the n

** **

9n

** **

**Step 2: **Subtract the common difference found in **Step 1 **from the first term in the sequence.

7-9=-2

**Step 3: **Put the algebraic term in **Step 1** and the number found in **Step 2** together.

9n-2

So, what was the point of this, why does this help us? Well, if we wanted to find the 127^{th} term, we’d have to write out all the numbers until we go to it. Having an algebraic expression lets us find the value of term without having to write them out.

*What is the 127 ^{th} term in the sequence given by 9n-2?*

We know that n represents the term in the sequence, so all we need to do is substitute in 127 for n!

9\times127-2=1143-2=1141

And this is our 127^{th} term!

## Example Questions

**Question 1:** Find the algebraic expression for the following sequence

7 10 13 16 19 22

**Step 1: **Find how much the sequences changes by each time, this is usually referred to as the common difference.

22-19=3

13-10=3

The common difference here is 3, and this goes in front of the n

** **

3n

** **

**Step 2: **Subtract the common difference found in **Step 1 **from the first term in the sequence.

7-3=4

**Step 3: **Put the algebraic term in **Step 1** and the number found in **Step 2** together.

3n+4

**Question 2:** What is the 17^{th} term in the arithmetic sequence given by 5n+3?

17\times 5+3=85+3=88