Free Nth Term Calculator
Find the formula for any sequence. Enter your terms and get the nth term formula with full step-by-step working. Perfect for GCSE and A-Level maths.
Understanding Sequence Types
Arithmetic Sequences
Each term differs from the previous by a constant value called the common difference (d).
Example:
2, 5, 8, 11, 14, ...
d = 3, T(n) = 3n - 1
Geometric Sequences
Each term is multiplied by a constant value called the common ratio (r).
Example:
3, 6, 12, 24, 48, ...
r = 2, T(n) = 3 × 2^(n-1)
Quadratic Sequences
The second differences are constant. The formula contains n².
Example:
1, 4, 9, 16, 25, ...
2nd diff = 2, T(n) = n²
Worked Examples
Finding the nth term of 5, 9, 13, 17, 21
- 1Find differences: 9-5=4, 13-9=4, 17-13=4, 21-17=4
- 2Constant difference: d = 4 → Arithmetic sequence
- 3Formula: T(n) = dn + (a-d) = 4n + (5-4) = 4n + 1
- 4Check: T(1) = 4(1)+1 = 5 ✓, T(2) = 4(2)+1 = 9 ✓
Finding the nth term of 3, 8, 15, 24, 35
- 1First differences: 5, 7, 9, 11 (not constant)
- 2Second differences: 2, 2, 2 → Quadratic!
- 3Find a: a = 2nd diff ÷ 2 = 2 ÷ 2 = 1
- 4Solve for b, c: T(n) = n² + 2n (using substitution)
GCSE Exam Tips for Sequences
Always Check Your Formula
Substitute n=1, n=2, n=3 back into your formula to verify it gives the original terms.
Remember (n-1) Not n
In T(n) = a + (n-1)d, the power is (n-1) because the first term is when n=1.
Show Your Differences
Always show the calculation of differences. Even if you spot the pattern, showing working earns marks.
Check if Value is in Sequence
Set T(n) = value and solve for n. If n is a positive integer, the value is in the sequence.
Frequently Asked Questions
How do I find the nth term of an arithmetic sequence?▼
For arithmetic sequences, use T(n) = a + (n-1)d, which simplifies to T(n) = dn + (a-d). Here, a is the first term and d is the common difference. For example, with sequence 3, 7, 11, 15: d = 4, a = 3, so T(n) = 4n + (3-4) = 4n - 1.
How do I find the nth term of a quadratic sequence?▼
For quadratic sequences, first find the second differences. The coefficient a = (second difference) ÷ 2. Then solve for b and c using the first terms. The formula is T(n) = an² + bn + c. Our calculator does this automatically!
What's the difference between arithmetic and geometric sequences?▼
An arithmetic sequence has a constant difference between consecutive terms (e.g., 2, 5, 8, 11 has common difference 3). A geometric sequence has a constant ratio between consecutive terms (e.g., 2, 6, 18, 54 has common ratio 3). Arithmetic sequences grow linearly; geometric sequences grow exponentially.
How do I find the sum of an arithmetic sequence?▼
Use the formula S = n/2 × (first term + last term), or S = n/2 × (2a + (n-1)d). For example, to sum the first 10 terms of 2, 5, 8, 11...: the 10th term is 29, so S = 10/2 × (2 + 29) = 5 × 31 = 155.
How can I check if a value is in a sequence?▼
Set the nth term formula equal to your target value and solve for n. If n is a positive whole number, the value is in the sequence at that position. If n is negative, fractional, or doesn't exist, the value is not in the sequence.
Explore More Free Tools
All our tools are 100% free with step-by-step learning