Finding the nth Term of a Linear Sequence
A linear sequence is a list of numbers with a constant difference between each successive term. Finding the “nth term” means finding a powerful formula that lets you calculate any term in the sequence without having to count up from the start.
Understanding the nth Term Formula
The Formula: $an + b$
The nth term of any linear sequence can be written in this form.
- $a$ is the common difference.
- $n$ is the term number (position).
- $b$ is the zero term (the adjustment needed to match the sequence).
What is a Linear Sequence?
It’s a sequence where the difference between terms is always the same. This is called the common difference.
How to Find the nth Term
Step 1: Find ‘a’
Find the common difference between the terms. This value is your $a$.
Step 2: Compare to Times Table
Compare your sequence to the ‘$a$’ times table.
Step 3: Find ‘b’
Work out the constant adjustment needed to get from the times table to your sequence. This is your $b$.
Step 4: Write the Rule
Combine your values to write the final rule in the form $an+b$.
Worked Example
Find the nth term of 5, 8, 11, 14, …
- Find the common difference (a): The sequence goes up by 3 each time, so $a=3$. This means the rule starts with $3n$.
- Compare to the 3 times table:
Position (n) 1 2 3 4 Our Sequence 5 8 11 14 3n (3 times table) 3 6 9 12 - Find the adjustment (b): To get from the $3n$ row to our sequence, we always need to add 2. So, $b=2$.
Answer: The nth term is $3n + 2$.
Tutor Insights
🤔 Common Misunderstandings
- Confusing the term number ($n$) with the value of the term.
- Forgetting the ‘b’ part. Just writing $3n$ for the sequence 5, 8, 11… is a common slip. The adjustment is crucial.
- Errors with negative differences. If a sequence is decreasing, remember that ‘a’ will be a negative number.
📝 Common Exam Mistakes
- Sign errors when calculating the adjustment (‘b’) for a decreasing sequence.
- Simple arithmetic mistakes when calculating the common difference or checking the rule.
- Incorrectly substituting $n$ when asked to find a specific term.
Practice Questions
- Find the nth term of the sequence: 4, 7, 10, 13, 16…
- A sequence has the nth term $6n – 2$. Is 58 a term in this sequence? Show your working.
- Find the nth term of the sequence: 50, 47, 44, 41, 38…
Show Answers
- Working: Common difference is 3 ($a=3$). The 3 times table starts with 3, but our sequence starts with 4, so we need to add 1 ($b=1$).
Answer: $3n+1$. - Working: Set $6n-2=58 \implies 6n=60 \implies n=10$. Since $n=10$ is a positive whole number, the answer is yes.
Answer: Yes, it is the 10th term. - Working: Common difference is -3 ($a=-3$). The -3 times table starts with -3, but our sequence starts with 50. The adjustment is $50 – (-3) = 53$ ($b=53$).
Answer: $-3n+53$ or $53-3n$.
