## 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 127th 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 127th 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 127th term!

## Example Questions

#### Question 1: Find the algebraic expression for the following sequence7          10        13        16        19        22

Answer

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 17th term in the arithmetic sequence given by $5n+3$?

Answer

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