# The nth Term of a Linear Sequence

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

Example 1
Consider the sequence beginning 3, 7, 11, 15, 19
a. Find the 𝑛th term of the sequence.
b. Find the 50th term in the sequence.

a. To find the coefficient of 𝑛, work out the term-to-term difference (how the sequence changes from one term to the next). \begin{aligned} &&+4&&+4&&+4&&+4\\&3&&7&&11&&15&&19\end{aligned}

This sequence is increasing by 4 each time so the 𝑛th term will start 4𝑛.

To find the ‘0th’ 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 𝑛th term is 4𝑛 – 1

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

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

4𝑛 – 1 = 4(50) – 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, …

First, find the 𝑛th term. The term-to-term difference is +3 and the 0th term is +2 so the 𝑛th term is 3𝑛 + 2

As well as using the 𝑛th 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 𝑛th term and form an equation to solve.

3𝑛 + 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. \begin{aligned} 3n+2&=43\\3n&=41\\n&=\frac{41}{3}\end{aligned}

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

Test yourself on Linear Sequences.

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