Number Sequence Calculator
Three sequence types that cover most of what school and finance actually use. The nth term and the sum are both given in closed form, so you do not have to count your way there.
The Sequence
The step field means difference for arithmetic and ratio for geometric.
Arithmetic: adding the same thing
An arithmetic sequence adds a constant difference each step, so the nth term is an = a1 + (n-1)d. Note the n minus one: the first term has had the difference applied zero times, and off-by-one here is the most common error in the topic. The sum of the first n terms is n times the average of the first and last, which is the trick Gauss is supposed to have used as a schoolboy to add the numbers one to a hundred in seconds — pair them from the outside in and every pair totals 101.
Geometric: multiplying by the same thing
A geometric sequence multiplies by a constant ratio, giving an = a1rn-1. This is the mathematics of compound interest, population growth and radioactive decay, and it behaves very differently depending on the ratio. With r greater than one the terms grow without bound; with r between minus one and one they shrink toward zero and the infinite sum converges to a1/(1-r); with r negative the terms alternate in sign. That convergence result is why an infinitely repeating decimal has a finite value.
Fibonacci: adding the two before
Each term is the sum of the previous two. It is neither arithmetic nor geometric, but the ratio of consecutive terms converges quickly to the golden ratio, about 1.6180 — by the tenth term it is already accurate to three decimal places. The sequence appears in phyllotaxis, in the branching of some plants and in the analysis of certain algorithms, though it is also attributed to far more natural phenomena than it actually explains.
Why closed forms matter
You could always generate terms one at a time until you reach the one you want, and for the tenth term that is fine. For the ten-thousandth it is not, and for a symbolic argument it is useless. A closed form gives the answer in one step and, more importantly, makes the structure visible: seeing rn-1 in the geometric formula tells you immediately that the growth is exponential in n, which no amount of listing terms would make obvious.
Where these show up in money
A fixed annual deposit is an arithmetic progression of contributions; the balance under compound interest is geometric. Annuity formulas are just the sum of a geometric series rearranged, which is why the same expression with r as one plus the interest rate appears in loan and pension mathematics. Recognizing the series underneath makes those formulas something you can reconstruct rather than memorise.
Frequently Asked Questions
Why is the nth term formula n minus 1?
When does an infinite geometric series have a finite sum?
Why does the Fibonacci ratio approach 1.618?
Can the common difference be negative?
Sources
Official publications only. Links open the original document in a new tab.
- National Institute of Standards and Technology Special Publication 811 — Guide for the use of the SI Significant figures and rounding conventions
- National Institute of Standards and Technology CODATA fundamental physical constants Values of the fundamental physical constants