The year is 1786. A schoolteacher in Brunswick, Germany, tells his class to add up the numbers 1 through 100. Keep them busy for a while, he thinks. A boy named Carl Friedrich Gauss walks up with the answer in under a minute: 5,050.
His trick? Pair the first and last terms. 1 + 100 = 101. 2 + 99 = 101. 3 + 98 = 101. Fifty pairs, each summing to 101. Total: 50 x 101 = 5,050. No addition needed - just multiplication. That is the arithmetic sequence sum formula, and it works for any evenly spaced list of numbers.
Two centuries later, the same math powers your savings plan (add $50 more each month), your compound interest (multiply by 1.08 each year), and the salary statistics that governments publish (median, not mean, because one CEO distorts the average for everyone).
Three tools. Three angles on sequences and central tendency.
The Gauss formula: arithmetic sequences
An arithmetic sequence adds the same amount each step. The natural numbers 1, 2, 3... add 1. A savings plan starting at $200 and adding $50/month gives 200, 250, 300, 350... The common difference d stays constant.
Two formulas handle everything:
| Formula | What it computes |
|---|---|
| a_n = a_1 + (n-1) x d | Any specific term |
| S_n = n x (a_1 + a_n) / 2 | Sum of first n terms |
The sum formula is Gauss's insight generalized. Pair first with last, multiply by count, divide by two.
Real example: You save $200 in month 1, increasing by $50 each month. After 12 months, your last deposit is a_12 = 200 + 11 x 50 = $750. Total saved: S_12 = 12 x (200 + 750) / 2 = $5,700. Not bad for a formula that a 10-year-old could understand.
The Arithmetic Sequence Calculator handles this instantly. Enter a_1, d, and n - get the term and sum with formulas shown step by step.
When adding is not enough: geometric sequences
An arithmetic sequence adds. A geometric sequence multiplies. Instead of d, you have a ratio r. Each term is the previous one times r.
The difference is dramatic. An arithmetic sequence with a_1 = 1 and d = 2 gives: 1, 3, 5, 7, 9, 11... After 20 terms, the last is 39. A geometric sequence with a_1 = 1 and r = 2 gives: 1, 2, 4, 8, 16, 32... After 20 terms, the last is 524,288.
That is the power - and danger - of exponential growth.
| Ratio r | Behavior | Real-world example |
|---|---|---|
| r = 1.08 | 8% compound growth | Retirement fund at 8% return |
| r = 2 | Doubling | Bacteria dividing every hour |
| r = 0.5 | Halving (decay) | Radioactive half-life |
| r = 0.85 | 15% annual loss | Car depreciation |
Compound interest example: $10,000 at 8% annual return. After 20 years: 10000 x 1.08^19 = $43,157. Your money quadrupled without you touching it. After 30 years: $93,173. After 40: $201,253. The curve gets steeper the longer you wait.
The Geometric Sequence Calculator models any multiplicative pattern. Enter first term, ratio, position - get the term value and partial sum.
Why your salary is not what the average says
Now a different kind of number: the median.
Take five households earning $35k, $42k, $48k, $52k, and $180k. The mean income is $71,400. The median is $48,000. Which better represents the "typical" household?
The median. Always. For skewed data.
The mean gets pulled toward the $180k outlier. The median sits in the exact middle - two households above, two below - and ignores extremes entirely. That is why governments report median income, not average income. And why real estate uses median house price, not average.
| Measure | 5 households above | Result |
|---|---|---|
| Mean | $35k, $42k, $48k, $52k, $180k | $71,400 |
| Median | Same data | $48,000 |
| Difference | $23,400 (the mean is 49% higher!) |
One $180k earner inflated the mean by nearly half. The median did not flinch.
How it works: sort the data. If the count is odd, pick the middle value. If even, average the two middle values. 
The Median Calculator does this instantly for any data set.
When to use which
These three tools answer different questions:
| Question | Tool | Why |
|---|---|---|
| "What is term #50 of 3, 7, 11, 15...?" | Arithmetic Sequence | Constant addition pattern |
| "How much will $5,000 grow at 6% for 25 years?" | Geometric Sequence | Compound (multiplicative) growth |
| "What is the typical value in a skewed data set?" | Median | Outlier-resistant central tendency |
Arithmetic for linear patterns. Geometric for exponential ones. Median for dirty data. The three together cover a surprising amount of real math.
Tools discussed in this article
- Arithmetic Sequence Calculator - find the nth term and Gauss sum of any evenly-spaced sequence
- Geometric Sequence Calculator - model exponential growth, decay and compound interest patterns
- Median Calculator - find the middle value of any data set, resistant to outlier distortion
More math tools
- Average Calculator
- Standard Deviation Calculator
- Linear Function Calculator
- Quadratic Function Calculator
- Powers & Roots Calculator
- Percentage Calculator
- Proportion Calculator
- Prime Factorization Calculator
- Factorial Calculator
- Logarithm Calculator
Calculate for specific term counts
- Arithmetic sequence 10 terms - standard problem
- Arithmetic sequence 20 terms - longer sequence
- Arithmetic sequence 100 terms - large sum
- Geometric sequence 10 terms - standard problem
- Geometric sequence 20 terms - longer sequence