When the steps never change
You start a savings plan with 200 in January and add 50 every month. A theater has 20 seats in the first row and 4 more in each row behind it. A contractor quotes a job at 500 for the first hour and 80 for every hour after. Each of these is an arithmetic sequence: a list where every term grows by the same fixed step, called the common difference.
This calculator finds any term in such a sequence without you writing out the whole list. Give it the first term, the common difference, and which position you want, and it returns the value, with the option to sum the first n terms as well.
The formula
Formula: a_n = a_1 + (n - 1) x d, where a_1 is the first term, d is the common difference, and n is the position you want.
The logic is simple counting. The first term needs zero steps, the second needs one, the third needs two, so the nth term needs (n - 1) steps of size d. The (n - 1) is the part people forget, and it matters: using n instead of (n - 1) overshoots every answer by exactly one step. The sum of the first n terms has its own shortcut: S_n = n / 2 x (a_1 + a_n), the count of terms halved, times the average of the first and last term.
How to use it
- Enter the first term of your sequence.
- Enter the common difference. Use a negative number for a decreasing sequence.
- Enter n, the position of the term you want.
- Read the nth term, and the sum of the first n terms if you need it.
Worked example
First term 5, common difference 3, and you want the 10th term. Substitute into the formula: a_10 = 5 + (10 - 1) x 3 = 5 + 9 x 3 = 5 + 27 = 32. Check by feel: the sequence runs 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, and the 10th value is indeed 32. The sum of the first ten terms: S_10 = 10 / 2 x (5 + 32) = 5 x 37 = 185.
Where steady steps show up
Loan and rent schedules with fixed monthly changes are arithmetic sequences in disguise, as are tiered pricing tables and production quotas that rise by a set amount each week. Teachers use them for grading curves and seating plans. If your data points are coordinates instead of a list, the distance formula calculator measures gaps between them; the mean calculator averages any batch of terms, and the GCF calculator simplifies fractions that show up in term ratios.
Common mistakes
- The off-by-one step: writing a_n = a_1 + n x d is the single most common error. Always use (n - 1).
- Sign of the difference: a sequence falling by 4 each time has d = -4, not 4. Dropping the minus flips the whole direction.
- Assuming every pattern is arithmetic: 2, 4, 8, 16 doubles each time, which is geometric, not arithmetic. Check that the gaps are equal before using the formula.
An honest note
This formula only fits perfectly straight-line growth. Real savings earn compounding interest and real costs rarely rise in exact steps, so treat sequence math as the clean backbone of the problem and layer the messy reality on top.