Arithmetic Series Sum Calculator
Find the sum of an arithmetic series from the first term, common difference, and number of terms, or from the first term, last term, and number of terms.
Arithmetic series means adding an arithmetic sequence
An arithmetic series is what you get when you add the terms of an arithmetic sequence. If the sequence is 5, 8, 11, 14, the arithmetic series is 5 + 8 + 11 + 14. This page targets both quick calculator intent and informational searches like how to find the sum of an arithmetic series, arithmetic series formula, and sum of first n terms of an arithmetic sequence.
OpenStax derives the arithmetic series formula by pairing the first and last terms in its arithmetic sequences and series explanation. That pairing idea is useful because every first-last pair has the same total.
Arithmetic series formulas
The second formula is usually the easiest when the last term is known. The first formula is better when the problem gives the common difference but not the last term.
Worked example: sum of the first 30 terms
The shortcut works because the first and last terms average to the same value as the second and second-last terms, the third and third-last terms, and so on. Instead of adding 30 numbers one by one, you multiply the average term by the number of terms.
When students confuse the nth term and the sum
If a problem asks for “the 30th term,” it wants a₃₀. If it asks for “the sum of the first 30 terms,” it wants S₃₀. Those are completely different outputs. In the example above, the 30th term is only 122, but the sum of the first 30 terms is 1920. Always look for words like term, total, sum, series, and first n terms.
Common real-world uses
Arithmetic series appear when equal increases build over time: saving $10 more each week, adding rows of seats where each row has two more seats than the previous row, stacking objects in evenly increasing rows, or calculating repeated fixed changes. The calculator is not only for abstract formulas; it is a fast way to total any pattern where the amount changes by the same number each step.
Common questions
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Use S_n = n/2(a_1 + a_n) when you know the first term, last term, and number of terms. If the last term is not given, first find it with a_n = a_1 + (n - 1)d, or use S_n = n/2[2a_1 + (n - 1)d]. Both formulas give the same result. The best choice depends on what information the problem gives you.
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An arithmetic sequence is the ordered list of terms. An arithmetic series is the sum of those terms. For example, 2, 5, 8, 11 is a sequence. The series is 2 + 5 + 8 + 11. A sequence question may ask for a specific term, while a series question asks for a total after adding terms together.
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In an arithmetic series, the terms are evenly spaced. That means the first and last term average to the same value as the second and second-last term, and so on. The average of the whole list is therefore the average of the first and last terms. Multiplying that average by the number of terms gives the total sum.
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Yes. If the terms are mostly negative, the sum can be negative. For example, -5, -10, -15, -20 is an arithmetic sequence, and its series sum is negative. A series can also start positive and become negative if the common difference is negative. The formula still works as long as the terms follow a constant difference.
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If d = 0, every term is the same. The arithmetic series sum becomes n times the repeated term. For example, if every term is 7 and there are 12 terms, the sum is 84. The usual formula still works because the first and last terms are both 7, so S_n = n/2(7 + 7) = 7n.
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Use the a_1, a_n, n formula when the first term, last term, and number of terms are known. Use the a_1, d, n formula when the first term, common difference, and number of terms are known. If a problem gives only two terms from the sequence, find the common difference first, then use the appropriate series formula.
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Yes. The odd numbers 1, 3, 5, 7, 9 form an arithmetic sequence with common difference 2. Their sums form square numbers: the sum of the first n odd numbers is n^2. This is a special arithmetic series pattern that often appears in algebra and number-pattern lessons.
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Yes. Arithmetic series formulas work with decimals, fractions, negative numbers, and mixed values. The key condition is a constant common difference. When working by hand, fractions often give cleaner exact answers than rounded decimals. This calculator shows decimal-style results for speed, but the same algebra applies to exact fractional values.