This guide has everything you need on finding and using the πth term of a linear sequence. If youβre sitting a higher tier paper, you may want to move on to finding and using the πth term of quadratic sequences once you’ve mastered this.

**Example 1**

Consider the sequence beginning 3, 7, 11, 15, 19

a. Find the πth term of the sequence.

b. Find the 50th term in the sequence.

**a. **To find the coefficient of π, work out the term-to-term difference (how the sequence changes from one term to the next).

This sequence is increasing by 4 each time so the πth term will start 4π.

To find the β0thβ term, find what value would be immediately before the first term in the sequence. Since this sequence is increasing by 4 each time, the previous term would be 4 less than 3, so -1.

So, the complete πth term is **4π β 1**.

**b.** We can use the πth term to find any term we want in a sequence, by substituting its position in for π

We want the 50th term so we substitute in *π* = 50:

4π β 1 = 4(50) β 1

= 199

The 50th term in this sequence is 199.

**Example 2**

Determine whether the number 43 appears in the sequence starting 5, 8, 11, 14, 17, β¦

First, find the π*th* term. The term-to-term difference is +3 and the 0th term is +2 so the πth term is **3π + 2**.

As well as using the πth term to find a value from a position, we can test whether a value has a position in the sequence. We set the value equal to the πth term and form an equation to solve.

3π* *+ 2 = 43

If the solution is an integer (a whole number) then 43 does appear in the sequence. If the solution is not an integer then 43 does not appear in the sequence.

41 is not divisible by 3 so the solution will not be an integer. This means 43 does not appear in this sequence.

## Testing Your Knowledge of the nth Term of a Linear Sequence

Now that you know how to represent inequalities on a number line, you’re probably ready for more revision – you can find more of our blogs **here**! You can also **subscribe to Beyond** for access to thousands of secondary teaching resources. You can **sign up for a free account here** and take a look around **at our free resources** before you subscribe too.