Arithmetic Sequence Calculator
Find the nth term, common difference, explicit formula, recursive formula, first terms, and series sum for an arithmetic sequence. This page is designed for both quick answers and for understanding how arithmetic sequences work in homework problems.
What this arithmetic sequence calculator actually solves
This calculator is built for the common classroom problems where a sequence changes by the same amount every step. It can find the nth term, the common difference, the explicit formula, the recursive formula, a short list of terms, and the sum of the first n terms. That means it covers searches such as arithmetic sequence calculator, nth term of an arithmetic sequence, common difference calculator, and arithmetic series sum from a sequence without forcing the student to jump between separate tools.
In a standard arithmetic sequence, the difference between consecutive terms is constant. OpenStax explains this same idea in its College Algebra section on arithmetic sequences, where the nth term is written as a first term plus repeated copies of the common difference.
Arithmetic sequence formulas
The two sum formulas are equivalent. Use the first one when you know the first term, common difference, and number of terms. Use the second one when you already know the first and last terms. OpenStax gives the arithmetic series sum as one-half of the number of terms times the first-plus-last term in its Intermediate Algebra lesson on arithmetic sequences.
How to know it is arithmetic
Look at the gaps between terms. If the sequence is 6, 10, 14, 18, 22, the differences are +4, +4, +4, +4, so it is arithmetic. If the sequence is 80, 70, 60, 50, the common difference is -10, so it is still arithmetic even though the numbers go down. A negative common difference does not break the rule; it simply means the sequence decreases at a constant rate.
| Sequence | Differences | Arithmetic? | Reason |
|---|---|---|---|
| 3, 8, 13, 18 | +5, +5, +5 | Yes | Same value is added each time. |
| 20, 17, 14, 11 | -3, -3, -3 | Yes | Same value is subtracted each time. |
| 2, 4, 8, 16 | +2, +4, +8 | No | The difference changes; this one is geometric. |
Worked example: find the 25th term and the sum
This example also shows why sequence and series questions are different. The term a₂₅ is only one number in the pattern. The series sum S₂₅ adds all 25 terms together. Many students lose marks because they calculate the 25th term when the problem asked for the sum of the first 25 terms.
Common mistakes this calculator helps avoid
The biggest mistake is using n instead of n - 1 in the nth-term formula. If the first term is already a₁, then you only move forward n - 1 steps to reach aₙ. Another mistake is treating the common difference as always positive. A sequence like 50, 45, 40, 35 has d = -5, not +5. Finally, remember that the common difference is a subtraction between consecutive terms, while the common ratio belongs to geometric sequences.
Common questions
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Use a_n = a_1 + (n - 1)d. Start with the first term, then add the common difference one fewer time than the term number. For example, to find the 20th term, you do not add the difference 20 times; you add it 19 times because the first term is already counted. This is why the expression uses n - 1. If a_1 = 6 and d = 3, then a_20 = 6 + 19(3) = 63.
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The common difference is the fixed amount added to get from one term to the next. You find it by subtracting consecutive terms, such as d = a_2 - a_1 or d = a_5 - a_4. If every pair of consecutive terms gives the same difference, the sequence is arithmetic. If the differences change, the pattern is not arithmetic, even if the terms look like they are increasing in a regular way at first glance.
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Yes. An arithmetic sequence can increase, decrease, or stay constant. If the common difference is positive, the terms increase. If the common difference is negative, the terms decrease. If the common difference is zero, every term is the same. For example, 30, 25, 20, 15 is arithmetic because the same value, -5, is added each time.
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An arithmetic sequence is the list of terms, such as 4, 9, 14, 19. An arithmetic series is the sum of those terms, such as 4 + 9 + 14 + 19. The nth-term formula finds one specific term in the list. The series formula finds the total after adding many terms. When reading a word problem, watch for words like 'find the 20th term' versus 'find the sum of the first 20 terms.'
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If you know two terms, first find the common difference using d = (a_j - a_i) / (j - i). After that, work backward to find the first term using a_1 = a_i - (i - 1)d. Once you have a_1 and d, the explicit formula follows from a_n = a_1 + (n - 1)d. This is useful when the problem gives something like a_4 = 18 and a_12 = 50 instead of listing the first few terms.
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They are closely related. If you graph term number n on the x-axis and term value a_n on the y-axis, an arithmetic sequence forms points on a straight line. The common difference acts like the slope, and the first term controls the starting value. The main difference is that a sequence usually uses positive whole-number inputs such as n = 1, 2, 3, while a linear function may allow any real x-value.
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Because the first term is already the starting point. To reach the second term, you add the common difference once. To reach the third term, you add it twice. So to reach the nth term, you add it n - 1 times. Using nd would move one step too far and give the wrong answer for every term except in the special case where d = 0.
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Yes. The terms and common difference can be whole numbers, fractions, negative numbers, or decimals. For example, 1.5, 2.25, 3.0, 3.75 is arithmetic with d = 0.75. The same formulas still work. The only extra care is rounding: if your homework expects an exact fraction, keep the common difference as a fraction instead of rounding too early.