Arithmetic Sequence Formulas
Find the nth term and sum of an arithmetic sequence.
Formulas: aₙ = a₁ + (n-1)d and S = n/2 × (a₁ + aₙ) with worked examples.
The Formulas
nth term: aₙ = a₁ + (n - 1) × d
Sum of n terms: S = n / 2 × (a₁ + aₙ)
An arithmetic sequence is a list of numbers where each term increases (or decreases) by the same fixed amount.
That fixed amount is called the common difference.
Variables
| Symbol | Meaning |
|---|---|
| aₙ | The nth term in the sequence |
| a₁ | The first term in the sequence |
| n | The position of the term (1st, 2nd, 3rd, etc.) |
| d | The common difference between consecutive terms |
| S | The sum of the first n terms |
Example 1
Find the 20th term of the sequence: 3, 7, 11, 15, ...
a₁ = 3, d = 7 - 3 = 4, n = 20
aₙ = a₁ + (n - 1) × d
a₂₀ = 3 + (20 - 1) × 4 = 3 + 19 × 4 = 3 + 76
a₂₀ = 79
Example 2
Find the sum of the first 10 terms of: 5, 8, 11, 14, ...
a₁ = 5, d = 3, n = 10
First find a₁₀: a₁₀ = 5 + (10 - 1) × 3 = 5 + 27 = 32
S = n / 2 × (a₁ + aₙ) = 10 / 2 × (5 + 32)
S = 5 × 37
S = 185
When to Use It
Use arithmetic sequence formulas when:
- Finding a specific term in a pattern that increases by a constant amount
- Calculating the total of evenly spaced numbers (e.g., sum of 1 to 100)
- Working with regular payment schedules or linear growth
- Solving problems involving seats in rows, stacked objects, or numbered patterns
Key Notes
- nth term: aₙ = a₁ + (n−1)d: a₁ is the first term and d is the common difference (added each step). The sequence is linear — a graph of aₙ vs n gives a straight line with slope d. Negative d gives a decreasing sequence.
- Sum of n terms: Sₙ = n/2 × (a₁ + aₙ) = n/2 × (2a₁ + (n−1)d): Gauss famously computed the sum 1+2+…+100 = 100/2 × (1+100) = 5,050 as a child using the first form. Both forms are equivalent; choose the one with the most known quantities.
- Arithmetic vs geometric sequences: Arithmetic: add a constant each step (linear growth). Geometric: multiply by a constant each step (exponential growth). Compound interest is geometric; simple interest is arithmetic. The long-run difference is dramatic.
- Arithmetic mean: The average of an arithmetic sequence is its middle term: (a₁ + aₙ)/2. This is consistent with the sum formula — total = number of terms × average term.
- Applications: Arithmetic sequences model salary steps with fixed annual raises, depreciation under the straight-line method (value decreases by a fixed amount per year), staircase step counts, uniform acceleration (velocity as an arithmetic sequence of time steps), and simple interest accumulation.