Arithmetic Sequence Calculator
Give the arithmetic sequence calculator a first term and a common difference, or any two terms, and it finds the nth term, the sum, and both formulas, exactly.
Term 10
- Term 10 a_n = a1 + (n - 1)d = 3 + (10 - 1) x 4 = 39
- 39
- Sum of the first 10 terms S_n = n (a1 + a_n) / 2 = 10 x (3 + 39) / 2 = 210. Pair the first term with the last, the second with the second-to-last, and every pair has the same total.
- 210
- Explicit formula a1 + (n - 1)d multiplied out: 3 + 4n - 4.
- a_n = 4n - 1
- Recursive formula Each term from the one before it.
- a1 = 3, a_n = a_(n - 1) + 4
- Average of the terms The average of the first and last terms, which is why the sum is n times this.
- 21
- First 10 terms
- 3, 7, 11, 15, 19, 23, 27, 31, 35, 39
| n | a_n | Sum to n |
|---|---|---|
| 1 | 3 | 3 |
| 2 | 7 | 10 |
| 3 | 11 | 21 |
| 4 | 15 | 36 |
| 5 | 19 | 55 |
| 6 | 23 | 78 |
| 7 | 27 | 105 |
| 8 | 31 | 136 |
| 9 | 35 | 171 |
| 10 | 39 | 210 |
- Every value is exact: fractions are kept as fractions and whole numbers of any length are computed in full. Past 40 digits a value is shown in scientific notation with its length.
How the Arithmetic Sequence Calculator works
An arithmetic sequence adds the same number each time: 3, 7, 11, 15, ... adds 4. That number is the common difference d, and with the first term a1 it fixes every term of the sequence.
The nth term. To reach term n you add d a total of n - 1 times, so
a_n = a1 + (n - 1)d
For a1 = 3 and d = 4, the 10th term is 3 + 9 x 4 = 39. Multiplied out, the formula is a_n = 4n - 1, the explicit formula, which gives any term directly from its position. The recursive formula says the same thing one step at a time: a1 = 3, and each term is the one before plus 4.
The sum of n terms. Write the sum forward and backward and add the two lines. Every column holds a first-plus-last pair with the same total, a1 + a_n, and there are n columns, so twice the sum is n(a1 + a_n):
S_n = n (a1 + a_n) / 2
For the first 10 terms of 3, 7, 11, ... that is 10 x (3 + 39) / 2 = 210. Adding up the terms of a sequence is called a series, which is why the same tool is also searched for as an arithmetic series calculator. The story that Gauss added 1 to 100 this way as a schoolboy, getting 100 x 101 / 2 = 5,050, is the same formula.
From two terms. If you know two terms but not the start, the common difference is the change in value over the change in position. Term 3 is 11 and term 8 is 31, so d = (31 - 11) / (8 - 3) = 4, and stepping back from term 3 gives a1 = 11 - 2 x 4 = 3. Choose two terms to see this worked through for any pair.
Geometric sequences multiply instead of add: 3, 6, 12, 24, ... has a common ratio r = 2. The nth term is a_n = a1 r^(n - 1) and the sum of n terms is a1 (1 - r^n) / (1 - r). When r is between -1 and 1 the terms shrink toward zero and the whole infinite series has a finite sum, a1 / (1 - r). Switch the sequence to geometric to use those.
Exact arithmetic. Starting at 3 with a common difference of 1/3 gives 3, 10/3, 11/3, 4, 13/3, and those five terms sum to 55/3. This calculator keeps such values as fractions rather than rounding them. Terms are computed directly from the formula, so the billionth term costs no more than the 10th.
Arithmetic Sequence Calculator FAQ
What is the formula for the nth term of an arithmetic sequence?
a_n = a1 + (n - 1)d, where a1 is the first term and d the common difference. For 3, 7, 11, ... the 10th term is 3 + 9 x 4 = 39.
How do you find the sum of an arithmetic sequence?
S_n = n (a1 + a_n) / 2: the number of terms times the average of the first and last. The first 10 terms of 3, 7, 11, ... sum to 10 x (3 + 39) / 2 = 210.
How do you find the common difference from two terms?
Divide the difference in the values by the difference in the positions. If term 3 is 11 and term 8 is 31, d = (31 - 11) / (8 - 3) = 4.
What is the difference between an arithmetic and a geometric sequence?
An arithmetic sequence adds the same number each time (3, 7, 11, 15); a geometric sequence multiplies by the same number each time (3, 6, 12, 24). Arithmetic sequences grow in a straight line, geometric ones exponentially.
Can the common difference be negative or a fraction?
Yes. A negative difference makes the sequence decrease (20, 17, 14, ...), and a fractional one gives fractional terms, which this calculator keeps exact.