Free Nth Term Calculator
Find the formula for any sequence with full step-by-step working. Identify arithmetic, geometric, and quadratic sequences, calculate specific terms, find partial sums, and compute the sum to infinity for convergent geometric series. 6 calculation modes, interactive learning, and rich visualisations. Perfect for GCSE and A-Level maths.
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What is a Sequence?
A sequence is an ordered list of numbers that follows a specific pattern or rule. Each number in the list is called a term, and the position of a term is denoted by n. For example, in the sequence 2, 5, 8, 11, 14, the first term T(1) = 2, the second term T(2) = 5, and so on.
A series is the sum of the terms in a sequence. So while 2, 5, 8, 11 is a sequence, 2 + 5 + 8 + 11 = 26 is a series. This distinction matters in GCSE and A-Level exams.
Notation
T(n) or an — the nth term. T(1) is the first term, also written as a. The common difference is d, the common ratio is r.
Finite vs Infinite
A finite sequence has a set number of terms (e.g., the first 10 even numbers). An infinite sequence continues forever, written with “...” (e.g., 1, 2, 3, 4, ...).
Types of Sequences
Arithmetic
Each term differs by a constant common difference (d). Graph: straight line.
Formula:
T(n) = a + (n-1)d
2, 5, 8, 11, 14, 17, 20, 23
Geometric
Each term is multiplied by a constant common ratio (r). Graph: exponential curve.
Formula:
T(n) = arn-1
3, 6, 12, 24, 48, 96, 192, 384
Quadratic
Second differences are constant. Graph: parabolic curve.
Formula:
T(n) = an² + bn + c
1, 4, 9, 16, 25, 36, 49, 64
Fibonacci
Each term is the sum of the previous two. Appears in nature (spirals, petals).
Rule:
T(n) = T(n-1) + T(n-2)
1, 1, 2, 3, 5, 8, 13, 21
Triangular
The nth triangular number: sum of integers 1 to n. Forms triangular dot patterns.
Formula:
T(n) = n(n+1)/2
1, 3, 6, 10, 15, 21, 28, 36
Arithmetic vs Geometric vs Quadratic
| Property | Arithmetic | Geometric | Quadratic |
|---|---|---|---|
| How to identify | Constant 1st differences | Constant ratio | Constant 2nd differences |
| nth term formula | T(n) = dn + (a-d) | T(n) = ar^(n-1) | T(n) = an²+bn+c |
| Sum formula | S = n/2(a+l) | S = a(rⁿ-1)/(r-1) | No simple formula |
| Growth type | Linear | Exponential | Polynomial |
| Graph shape | Straight line | Exponential curve | Parabola |
| Exam level | GCSE + A-Level | GCSE + A-Level | GCSE (Higher) |
Sum Formulas
Arithmetic Sum
Sₙ = n/2 × (a + l)
Sₙ = n/2 × (2a + (n-1)d)
Where a = first term, l = last term, d = common difference, n = number of terms. Derived by pairing terms from opposite ends: each pair sums to (a + l).
Geometric Sum
Sₙ = a(rⁿ - 1) / (r - 1)
Sₙ = a(1 - rⁿ) / (1 - r)
Use the first form when r > 1, the second when r < 1. Both are equivalent. Not valid when r = 1 (just use S = na).
Infinite Geometric Sum
S∞ = a / (1 - r)
Only valid when |r| < 1. As n → ∞, rⁿ → 0, so the partial sum formula simplifies to a/(1-r). This is an A-Level topic covered by all exam boards.
Convergence and Divergence
A geometric series converges (has a finite sum to infinity) when the absolute value of the common ratio is less than 1: |r| < 1. This means each term gets progressively smaller, so the total sum approaches a fixed value.
When |r| ≥ 1, the series diverges — the terms don't shrink (or grow), so the sum increases without bound.
Convergent Example
½ + ¼ + ⅛ + 1/16 + …
Here a = ½, r = ½, |r| = 0.5 < 1
S∞ = 0.5 / (1 - 0.5) = 1
Divergent Example
2 + 6 + 18 + 54 + …
Here a = 2, r = 3, |r| = 3 ≥ 1
No finite sum — series diverges
Special Sequences Reference
| Sequence | Formula | First 8 Terms |
|---|---|---|
| Triangular | n(n+1)/2 | 1, 3, 6, 10, 15, 21, 28, 36 |
| Square | n² | 1, 4, 9, 16, 25, 36, 49, 64 |
| Cube | n³ | 1, 8, 27, 64, 125, 216, 343, 512 |
| Fibonacci | T(n-1)+T(n-2) | 1, 1, 2, 3, 5, 8, 13, 21 |
| Primes | No formula | 2, 3, 5, 7, 11, 13, 17, 19 |
| Powers of 2 | 2ⁿ | 2, 4, 8, 16, 32, 64, 128, 256 |
Common Mistakes to Avoid
✗Using n instead of (n-1) in T(n) = a + nd
✓T(n) = a + (n-1)d — the first term is at n = 1, not n = 0
✗Confusing difference (arithmetic) with ratio (geometric)
✓Arithmetic: subtract consecutive terms. Geometric: divide consecutive terms
✗Forgetting to check second differences for quadratic
✓If 1st diffs are not constant, always check 2nd diffs before trying ratios
✗Using a = 2nd difference for the quadratic coefficient
✓a = (2nd difference) ÷ 2. The ÷ 2 is essential!
✗Computing S∞ when |r| ≥ 1
✓S∞ = a/(1-r) only works when |r| < 1. Check convergence first!
✗Confusing Sₙ (partial sum) with S∞ (infinite sum)
✓Sₙ = a(1-rⁿ)/(1-r) gives the sum of n terms. S∞ = a/(1-r) is the limit
✗Getting the sign wrong on the constant c in T(n) = dn + c
✓c = a - d (first term minus common difference), not just a
✗Not verifying the formula by substituting n = 1, 2, 3
✓Always check: plug n = 1 into your formula — does it give the first term?
Worked Examples
Find 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) = 4n + (5-4) = 4n + 1
- 4Check: T(1) = 4(1)+1 = 5 ✓, T(2) = 4(2)+1 = 9 ✓
Find 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 = 2 ÷ 2 = 1
- 4Solve: T(n) = n² + 2n
Sum of first 20 terms with r = 0.8
- 1Given: a = 10, r = 0.8, n = 20
- 2Formula: Sₙ = a(1 - rⁿ) / (1 - r)
- 3Calculate: S₂₀ = 10(1 - 0.8²⁰) / (1 - 0.8) = 10(1 - 0.01153) / 0.2
- 4Result: S₂₀ ≈ 49.42
Prove ½ + ¼ + ⅛ + … = 1
- 1Identify: a = ½, r = ½, |r| = 0.5 < 1 → converges
- 2Apply: S∞ = a / (1 - r) = 0.5 / (1 - 0.5)
- 3Simplify: S∞ = 0.5 / 0.5 = 1 ✓
- 4Verify: Partial sums: 0.5, 0.75, 0.875, 0.9375, … → 1
Real-World Applications
Compound Interest
Savings and investments grow as geometric sequences. If you invest £1000 at 5% annual interest, the balance each year is 1000, 1050, 1102.50, … — a geometric sequence with r = 1.05.
Population Growth
Bacteria double every hour — a geometric sequence with r = 2. Understanding this helps model disease spread, wildlife populations, and resource depletion.
Patterns in Nature
The Fibonacci sequence appears in sunflower spirals, pinecone patterns, and nautilus shells. Leaf arrangement (phyllotaxis) often follows Fibonacci ratios approaching the golden ratio 1.618.
Scheduling & Planning
Seating plans in auditoriums often use arithmetic sequences (each row has 2 more seats than the last). Stacking problems and resource allocation frequently involve sum formulas.
How to Identify Any Sequence
Follow this decision flowchart to identify any sequence type:
Calculate the first differences
Subtract each term from the next: T(2)-T(1), T(3)-T(2), etc.
Are first differences constant?
Yes → Arithmetic sequence. The common difference is d.
No? Calculate the ratios T(n+1) ÷ T(n)
Constant ratio → Geometric sequence. The common ratio is r.
No? Calculate the second differences
Constant 2nd differences → Quadratic sequence. Use a = 2nd diff ÷ 2.
Still no pattern? Check special sequences
Try Fibonacci (each term = sum of previous two), or look up known sequences like triangular, square, cube numbers.
Frequently Asked Questions
How do I find the nth term of an arithmetic sequence?
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, 3, 7, 11, 15: d = 4, a = 3, so T(n) = 4n - 1.
How do I find the nth term of a quadratic sequence?
Find the second differences. The coefficient a = (2nd diff) ÷ 2. Then solve for b and c using T(1) and T(2). The formula is T(n) = an² + bn + c.
What's the difference between arithmetic and geometric sequences?
Arithmetic: constant difference between terms (linear growth). Geometric: constant ratio between terms (exponential growth). Check differences first, then ratios.
How do I find the sum of an arithmetic sequence?
Use S = n/2 × (first + last), or S = n/2 × (2a + (n-1)d). For the first 10 terms of 2, 5, 8...: 10th term = 29, S = 10/2 × (2+29) = 155.
What is the sum to infinity of a geometric series?
When |r| < 1, S∞ = a/(1-r). For example, 8 + 4 + 2 + 1 + ... has a = 8, r = 0.5, so S∞ = 8/0.5 = 16. If |r| ≥ 1, the series diverges.
When does a geometric series converge?
A geometric series converges when |r| < 1 (common ratio strictly between -1 and 1). The terms get smaller and smaller, approaching zero, so the sum approaches a finite limit.
What are Fibonacci numbers?
The Fibonacci sequence starts 1, 1, 2, 3, 5, 8, 13, 21... Each term is the sum of the two before it. It appears in nature (sunflower spirals) and the ratio of consecutive terms approaches the golden ratio φ ≈ 1.618.
How do I check if a value is in a sequence?
Set T(n) = your target value and solve for n. If n is a positive whole number, the value is in the sequence. If n is negative, fractional, or undefined, it is not.
What is a recurrence relation?
A recurrence relation defines each term using previous terms, like T(n+1) = T(n) + 3. This is equivalent to an explicit formula (here T(n) = 3n + c). Our calculator finds explicit formulas directly.
Which exam boards test sequences?
All UK exam boards (AQA, Edexcel, OCR) cover sequences at GCSE and A-Level. Arithmetic and quadratic are GCSE topics. Geometric series, convergence, and sum to infinity are A-Level topics.
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