Sum of Series Calculator
An arithmetic series that starts at 2 and adds 3 each step for 10 terms sums to 155. A sum of series calculator adds up every term in an arithmetic or geometric sequence without forcing you to list and add each value by hand. Pick the series type, enter the first term, the number of terms, and either the common difference (arithmetic) or the common ratio (geometric), and the calculator returns the total along with the value of the last term. An arithmetic series has a constant gap between consecutive terms, so 2, 5, 8, 11 keeps adding 3. A geometric series has a constant multiplier instead, so 2, 6, 18, 54 keeps multiplying by 3. The two series types use different formulas and grow at very different rates, which is why this calculator asks which mode you want before it asks for the numbers. Arithmetic sums grow at a steady, predictable pace, while geometric sums can grow slowly for a while and then explode once the ratio and term count both climb. This kind of series math shows up in algebra and precalculus coursework, in loan and investment problems that model repeated growth, and in any situation where a pattern repeats a fixed number of times with either a constant add-on or a constant multiplier. Students checking homework, teachers building answer keys, and anyone estimating compounding-style totals can use the same two formulas this calculator runs.
Quick answer
An arithmetic series sum uses sum = n/2 x (2a1 + (n-1)d), where a1 is the first term, d is the common difference, and n is the number of terms.
Sum of series
155
Last term
29
Series type
arithmetic
What this tells you
- •An arithmetic series sum uses sum = n/2 x (2a1 + (n-1)d), where a1 is the first term, d is the common difference, and n is the number of terms.
- •A geometric series sum uses sum = a1 x (1 - r^n) / (1 - r), where a1 is the first term, r is the common ratio, and n is the number of terms.
- •When the common ratio r equals 1 in a geometric series, the formula above divides by zero, so the calculator switches to sum = a1 x n instead, since every term is identical to a1 in that case.
- •The number of terms n must be a positive whole number for either series type. A fractional or negative term count has no meaningful sum.
- •A geometric series with a large ratio and a large number of terms can produce a sum too large to represent as a normal number. The calculator flags that case instead of returning a broken or infinite result.
- •The common difference in an arithmetic series and the common ratio in a geometric series can each be negative, zero, or a decimal, and the formulas still apply without changes.
How to Use
- 1Choose the series type first: arithmetic if consecutive terms share a constant difference, or geometric if they share a constant ratio.
- 2Enter the first term (a1), meaning the value of the very first number in the sequence.
- 3Enter the number of terms (n), meaning how many values in the sequence you want to add together.
- 4For an arithmetic series, enter the common difference (d), the fixed amount added to get from one term to the next. For a geometric series, enter the common ratio (r), the fixed multiplier between terms.
- 5Read the result for the total sum plus the value of the last term in the sequence, which is useful for checking your work against the term list directly.
How It Works
Formula
Arithmetic: sum = n/2 x (2a1 + (n-1)d). Geometric: sum = a1 x (1 - r^n) / (1 - r) for r != 1, or a1 x n when r = 1.For an arithmetic series, a1 is the first term, d is the common difference between consecutive terms, and n is the number of terms being summed. The formula works by pairing the first and last terms, the second and second-to-last terms, and so on, and noticing every pair adds up to the same value, then multiplying that pair sum by half the number of terms. For example, with a1 = 2, d = 3, and n = 10, the sum is 10/2 x (2x2 + 9x3) = 5 x 31 = 155. For a geometric series, a1 is the first term, r is the common ratio between consecutive terms, and n is the number of terms. The standard formula divides by (1 - r), which is undefined when r equals 1, so the calculator uses the simpler sum = a1 x n in that special case, since a ratio of 1 means every term equals the first term. For example, with a1 = 3, r = 2, and n = 5, the sum is 3 x (1 - 2^5) / (1 - 2) = 3 x (1 - 32) / (-1) = 3 x 31 = 93.
Calculation note: values are processed in the order shown above, using the current input units.
Worked Examples
Arithmetic series with a positive difference
The sequence is 2, 5, 8, 11, 14, 17, 20, 23, 26, 29. Using sum = n/2 x (2a1 + (n-1)d) gives 10/2 x (4 + 27) = 5 x 31 = 155.
Geometric series with a ratio above 1
The sequence is 3, 6, 12, 24, 48. Using sum = a1 x (1 - r^n) / (1 - r) gives 3 x (1 - 32) / (1 - 2) = 3 x 31 = 93.
Geometric series with a ratio of exactly 1
Every term equals the first term when the ratio is 1, so the sequence is 7, 7, 7, 7. The sum is simply a1 x n = 7 x 4 = 28, avoiding the division by zero in the general formula.
Arithmetic series with a negative difference
The sequence is 20, 16, 12, 8, 4, 0. Using sum = n/2 x (2a1 + (n-1)d) gives 6/2 x (40 - 20) = 3 x 20 = 60.
Common mistakes
- Entering the common ratio into the common difference field, or the reverse. Arithmetic series need a common difference, and geometric series need a common ratio. Mixing them up produces a meaningless sum.
- Forgetting that the number of terms must be a positive whole number. Half a term or zero terms has no valid sum.
- Assuming a geometric series formula works when the ratio is exactly 1. The standard formula divides by (1 - r), which is undefined at r = 1, so that case needs the simpler sum = a1 x n instead.
- Not noticing that a geometric ratio less than -1, or greater than 1, combined with a large number of terms, can produce a sum too large to compute in a normal number range. The calculator returns a clear message instead of a silently broken result.
- Confusing the sum of the series with the value of the last term. The two results answer different questions, and this calculator reports both so you can check either one against your own work.