Arithmetic Sequence
nth term and sum of sequence
Arithmetic sequence: aₙ = a₁ + (n−1)d
About This Calculator
An arithmetic sequence is a list of numbers where each term differs from the previous by a constant amount called the common difference. These sequences appear in finance (regular deposits), physics (constant acceleration), and are foundational to series and calculus.
Formula
nth term: a_n = a_1 + (n-1)*d
Sum of n terms: S_n = n*(a_1 + a_n)/2 = n*(2*a_1 + (n-1)*d)/2
Common difference: d = a_(n+1) - a_n
Example Calculation
Sequence 3, 7, 11, 15, ... (a1=3, d=4)
- 10th term = 3 + (10-1)*4 = 3 + 36 = 39
- Sum of 10 terms = 10*(3+39)/2 = 10*42/2 = 210
10th term = 39; Sum of first 10 terms = 210
First 10 Terms for Different Common Differences (a1=1)
| Term | d=1 | d=2 | d=3 | d=5 |
|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 |
| 2 | 2 | 3 | 4 | 6 |
| 3 | 3 | 5 | 7 | 11 |
| 4 | 4 | 7 | 10 | 16 |
| 5 | 5 | 9 | 13 | 21 |
| 10 | 10 | 19 | 28 | 46 |
Frequently Asked Questions
What is the difference between arithmetic and geometric sequences?
An arithmetic sequence adds a constant (d) each term: 2, 5, 8, 11. A geometric sequence multiplies by a constant (r) each term: 2, 6, 18, 54. Arithmetic grows linearly; geometric grows exponentially.
How does Carl Gauss famously summed 1 to 100?
Gauss paired numbers from each end: 1+100=101, 2+99=101, ..., 50+51=101. There are 50 such pairs, so the sum is 50*101=5050. This generalizes to S_n = n*(a1+an)/2.
What is the sum of the first n natural numbers?
Using the formula S_n = n*(n+1)/2: sum of 1 to 100 = 100*101/2 = 5050; sum of 1 to n is always a triangular number.