Sequence Sum Calculator

Introduction

The formulas for arithmetic and geometric series are familiar, yet the edge cases still trip people up: a ratio of exactly one, or an odd number of terms when halving. This page writes out the last term, the sum, the mean and the general term, then recomputes the first half separately.

The example starts at 3 with a difference of 2 over 10 terms. The last term is 21, the sum 120, the mean 12, the general term a_n = 3 + (n−1) × 2, and with k at 5 the fifth term is 11 and the sum of the first five is 35. Switch to geometric with ratio 2 and the last term becomes 1536 with a sum of 3069.

A ratio of one would make the geometric sum denominator zero, so that case takes the a1 times n branch instead of returning NaN or infinity.

Key features

• Two series: The input is a difference or a ratio depending on type. A ratio of 1 is treated as constant.
• Sum of n terms: The closed-form sum is returned, with the step-by-step addition listed for checking.
• Last term: Both the last term and the general formula are shown, not just a single number.
• k-th term: Enter k for any single term. k above the total count is rejected.
• Partial sum: The cumulative sum up to k is included so it can be compared with the full sum.
• Mean: The average of the sequence has its own result cell.

How to use it

1. First term a1: Enter the first number of the sequence.
2. Difference / ratio: Arithmetic uses a common difference, geometric a common ratio; a ratio of 1 is treated as a constant sequence.
3. Number of terms n: Enter a positive whole number. Zero and negatives are refused.
4. Sequence type: Choose arithmetic or geometric; the general term changes with it.
5. Read the general term: Press Calculate and read the formula first, then the last term and sum of n terms. The k-th term and the sum of the first k terms use the k shown in the breakdown.

Things to keep in mind

• Difference or ratio: Arithmetic uses a common difference, geometric a ratio; ratio 1 is treated as constant.
• Term count: Term count must be a positive integer.
• The k-th term: k cannot exceed the total number of terms.
• General term: The general term is shown as a formula, not collapsed to a value.
• Large term counts: Very large term counts may hit integer limits.

The general term is written as a formula rather than collapsed into a number, because it is the piece you need when extending the sequence by hand. A geometric ratio of 1 is handled as a constant sequence, and the k-th term and partial sum use the k named in the breakdown. Very large term counts can outgrow ordinary integer arithmetic, so treat those outputs as indicative.

For an arithmetic sequence starting at 3 with a difference of 4 over 10 terms, the last term is 39 and the sum is 210; the mean of the terms sits between the first and last. Switching to geometric with a ratio of 2 grows both figures far faster, which is why the general term is shown as a formula instead of a single number.