An arithmetic sequence adds the same amount at every step. From a first term a and a common difference d, the nth term is a + (n − 1)d — you take n − 1 steps to get from the first term to the nth, not n. That off-by-one is the single most common slip with these sequences.
Summing one is easier than it looks. Pair the first term with the last, the second with the second-last, and so on: every pair adds to the same total. That gives the sum as n times the average of the first and last terms, the trick Gauss is said to have found as a schoolboy when told to add the numbers to a hundred. Because the mean term sits exactly in the middle, the sum is also n times the mean, which is why arithmetic sequences turn up in depreciation schedules and simple interest.
The formula
a- The first term
d- The common difference added at each step
n- Which term you want
aₙ- The value of that term
Sₙ- The sum of the first n terms
How it works, step by step
- Enter the first term of the sequence.
- Enter the common difference — negative for a falling sequence.
- Enter which term you want and read it on the gauge.
- The sum to that term and the mean appear below.
Worked examples
First term 5, difference 3
The twentieth term is 5 + 19 × 3 = 62. The sum of the first twenty is 20 × (5 + 62) ÷ 2 = 670, and the mean term is 33.5, sitting exactly halfway between the first and last.
Gauss and the first hundred integers
With a = 1 and d = 1, the hundredth term is 100 and the sum is 100 × (1 + 100) ÷ 2 = 5,050. Pairing 1 with 100, 2 with 99 and so on gives fifty pairs of 101, which is the same arithmetic seen a different way.
How to read your score
Frequently asked questions
Why is it n − 1 rather than n in the formula?
Because the first term needs no steps at all. Reaching the fifth term takes four additions of the common difference, so the multiplier is one less than the term number.
How does the sum formula work?
Pair the terms from the outside in. Each pair adds to the same total, the first term plus the last, and there are n ÷ 2 pairs. Multiplying gives n times the average of the ends.
What if the common difference is negative?
The sequence falls, and the terms eventually go below zero. The formulae are unchanged; the zero-crossing figure tells you at roughly which term the sign flips.
Where do arithmetic sequences appear in practice?
Straight-line depreciation, simple interest, seating rows in a theatre and any schedule that adds a fixed amount each period. Whenever growth is by a fixed amount rather than a fixed percentage, the sequence is arithmetic.
An arithmetic sequence with a = 5, d = 3
| n | Term aₙ | Sum Sₙ | Mean term |
|---|---|---|---|
| 1 | 5 | 5 | 5.0 |
| 2 | 8 | 13 | 6.5 |
| 5 | 17 | 55 | 11.0 |
| 10 | 32 | 185 | 18.5 |
| 20 | 62 | 670 | 33.5 |
| 50 | 152 | 3,925 | 78.5 |
| 100 | 302 | 15,350 | 153.5 |
The sum is always the number of terms multiplied by the mean of the first and last.