Algebra
Grade 3-5Sequences
A sequence is a list of numbers that follows a pattern, and being able to describe that pattern with an nth term formula makes it possible to find any term directly, without listing every term before it. This lesson covers finding the term-to-term rule, finding the nth term of a linear sequence, using the nth term to find a specific term, and checking whether a given number is a term in the sequence.
Written by Asad, Co-Founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Finding the term-to-term rule and next terms
Find the difference between consecutive terms to identify the term-to-term rule, then apply this rule to find further terms.
For the sequence 3, 7, 11, 15, ..., each term is 4 more than the last, so the term-to-term rule is "add 4", and the next two terms are 19 and 23.
Finding the nth term of a linear sequence
Find the common difference between consecutive terms, and use it as the coefficient of n. Then find the constant by subtracting the common difference from the first term.
For 3, 7, 11, 15, ..., the common difference is 4, so the nth term starts with 4n. Subtracting the common difference from the first term gives 3 − 4 = −1, so the nth term is 4n − 1.
Using the nth term to find a specific term
Substitute the position number into the nth term formula to find the value of that term.
The nth term of a sequence is 6n + 2. To find the 10th term, substitute n = 10: 6(10) + 2 = 62.
Checking whether a number is a term in the sequence
Set the nth term formula equal to the given number and solve for n. If n is a positive whole number, the given number is a term in the sequence; if not, it is not a term.
The nth term of a sequence is 3n + 1. Setting 3n + 1 = 100 gives n = 33, a whole number, so 100 is the 33rd term of the sequence.
Worked Examples
Three exam-style questions, fully solved.
Write the next two terms of the sequence 3, 7, 11, 15, ...
Easy- 1.Find the term-to-term rule: each term is 4 more than the last
- 2.Apply the rule to the last term: 15 + 4 = 19, then 19 + 4 = 23
Answer: 19, 23
Find the nth term of the sequence 3, 7, 11, 15, ...
Medium- 1.Find the common difference: 4
- 2.Subtract the common difference from the first term: 3 − 4 = −1
Answer: 4n − 1
The nth term of a sequence is 3n + 1. Is 100 a term in this sequence?
Hard- 1.Set the nth term equal to 100: 3n + 1 = 100
- 2.Solve for n: 3n = 99, so n = 33
- 3.Check whether n is a whole number: 33 is a whole number, so 100 is a term
Answer: Yes, 100 is the 33rd term
Avoid These
The most common mistakes students make.
Miscounting the common difference, especially in a decreasing sequence, forgetting to include the negative sign.
Using the wrong constant when writing the nth term formula, instead of correctly subtracting the common difference from the first term.
Substituting the wrong value of n when finding a specific term, such as using n = 9 or n = 11 instead of n = 10 for the 10th term.
When checking whether a number is a term in the sequence, forgetting to check whether the resulting value of n is a whole number, and instead just checking that an equation could be solved.
Confusing the term-to-term rule, which describes how to get from one term to the next, with the nth term formula, which gives any term directly from its position.
FAQ
Questions parents and students ask.
Before this topic, make sure you know
What to learn next
Want a plan built around your child specifically?
Get our free 8-video course, or book a free Roadmap Call for a personalised plan.