The formula
How it works
Find the nth term and the running total of an arithmetic or geometric sequence. Arithmetic sequences add a fixed amount each step; geometric ones multiply by a fixed ratio. Enter the first term, the step and which term you want.
FAQ
What’s the difference between the two sequences?
An arithmetic sequence changes by adding the same number each time (2, 5, 8, 11…). A geometric sequence changes by multiplying by the same number each time (2, 6, 18, 54…), so it grows much faster.
What is the “sum” shown?
It is the total of all terms from the first up to the nth — called a series. Both sequence types have a neat formula for this sum, so you do not have to add the terms one by one.
Can the common difference or ratio be negative?
Yes. A negative difference makes an arithmetic sequence count down, and a negative ratio makes a geometric sequence flip sign every term while its size still grows or shrinks.
What happens if the common ratio is 1?
Every term in the geometric sequence equals the first term, so the sum is just the first term multiplied by the number of terms — the usual sum formula cannot be used since it would divide by zero.
Does a geometric sequence always grow forever?
No. If the common ratio is between −1 and 1 the terms shrink toward zero instead of growing, and the sum of infinitely many terms converges to a finite value.
How can I check the result by hand?
List out the first few terms using the rule — add d each time for arithmetic, or multiply by r each time for geometric — and add them up; the running total should match the sum the calculator shows.
What is a real-life example of each type?
Arithmetic: a taxi fare that adds a fixed amount per kilometre. Geometric: a bank balance earning a fixed percentage of compound interest each year.
About the number sequence calculator
This calculator works with the two most common number sequences — arithmetic and geometric — finding any term you ask for and the sum of the sequence up to that point. A sequence is just an ordered list of numbers built by a rule, and these two rules cover a huge amount of maths: adding a fixed step, or multiplying by a fixed factor. The calculator handles both the individual term and the running total.
How to use it
Choose arithmetic or geometric, then enter the first term, the common difference (for arithmetic) or common ratio (for geometric), and the term number you want. The calculator returns that term, the sum up to it, and the first several terms so you can see the pattern. For example, an arithmetic sequence starting at 2 with a difference of 3 gives a 5th term of 14 and a sum of 40 (2 + 5 + 8 + 11 + 14).
The formula
For an arithmetic sequence the nth term is and the sum is . For a geometric sequence the nth term is and the sum is (for ). The arithmetic sum pairs terms from both ends, while the geometric sum uses the way each term is a multiple of the last.
Where it is used
Arithmetic sequences model anything that changes by a steady amount — saving a fixed sum each month, or seats increasing row by row in a theatre. Geometric sequences model anything that grows or shrinks by a percentage — compound interest, population growth, or radioactive decay. Series sums turn up in finance, physics and computer science, wherever you need the total of many regularly changing terms.