Arithmetic sequences, common difference, formulas
High School Essentials • 9-12
An arithmetic sequence is a list of numbers where each term is obtained by adding the same fixed value -- called the common difference -- to the previous term. These sequences model situations with constant, steady change: saving $50 per week, climbing stairs of equal height, or counting by threes.
A sequence is arithmetic if the difference between consecutive terms is always the same.
3, 7, 11, 15, 19, ... (d = 4)
20, 17, 14, 11, 8, ... (d = -3)
2, 4, 8, 16, 32, ... (differences double)
1, 1, 2, 3, 5, 8, ... (Fibonacci)
| Formula | Expression | Use |
|---|---|---|
| Explicit (nth term) | an = a1 + (n - 1)d | Jump directly to any term |
| Recursive | an = an-1 + d, with a1 given | Define each term from the previous one |
| Sum of n terms | Sn = n/2 · (a1 + an) | Total of the first n terms |
The sum formula works by pairing the first and last terms (which add to the same value), then multiplying by the number of pairs. Gauss famously used this trick as a child to sum 1 + 2 + 3 + ... + 100 = 100/2 × (1 + 100) = 5,050.
Find the 20th term of the sequence 5, 9, 13, 17, ...
An arithmetic sequence has a5 = 22 and a12 = 43. Find d and a1.
Find the sum of the first 30 terms of the sequence 2, 5, 8, 11, ...
In the formula an = a1 + (n - 1)d, note it is (n - 1), not n. The first term a1 uses zero steps of d. A frequent error is writing an = a1 + nd, which gives you the (n+1)th term instead.
a1 = -3, d = 4. a15 = -3 + 14(4) = -3 + 56 = 53.
a1 = 100, d = -7. an = 100 + (n - 1)(-7) = 107 - 7n.
an = 4 + (n - 1)(7) = 200. So 7(n - 1) = 196, n - 1 = 28, n = 29.
S200 = 200/2 × (1 + 200) = 100 × 201 = 20,100.
From a3 to a7: 4 steps, difference = 30 - 14 = 16, so d = 4. a1 = 14 - 2(4) = 6. a10 = 6 + 9(4) = 42.