Arithmetic Sequence Calculator

Recursive vs. Explicit Formula

There are actually two ways to define a sequence in mathematical terms. The first of this, which we demonstrated above, is referred to as a recursive formula. We define the arithmetic sequence calculations in a recursive formula based on the prior item. Take your starting value, add X, keep going until you hit the nth term. This defines the pattern for the series and doesn't require higher level math (creating formulas).

You can also define what is known as an explicit formula, where you don't need to iterate through the values of the arithmetic sequence to get a number. You simply enter the number of terms which you want, along with terms that describe how the arithmetic sequence is constructed, and the explicit formula will tell you the value of the nth term.

Don't underestimate the power of a recursive formula. It can tolerate more complexity than an explicit formula, since you're defining the second term as a function of the previous number. As long as things move at a constant value or ratio, you can manage the finite arithmetic progression.

Explicit Formula: The Arithmetic Sequence Formula

You can solve for the answer to the arithmetic sequence question above using algebra. The nth value of an arithmetic sequence can be calculated as:
Starting point + (n - 1) x common difference

So the 5 th term for a series starting at 3 (the initial term), with a common difference of 4, and where n = 5 would be:
3 + (5-1) x 4 = 19

This arithmetic sequence equation shows what happens as we add a successive term to the series (add the constant difference) or backtrack to the previous term. A quick way to understand a number sequence, without using an arithmetic series calculator.

Other Series

There are other common series out there. One is the geometric series, generated by multiplying each term by a constant. This shows the geometric progression of a variable. (we have a geometric series calculator as well; the geometric sequence calculator can be found here.) The geometric series equivalent of the common difference is known as the common ratio, which is the ratio between the prior term and the next term in the sequence (expressed as a multiple of the prior term).

Similar to the arithmetic sequence equation, the geometric series has a formula for directly calculating values: the geometric sequence formula. This is expressed as the starting term plus the sum of (starting term) * (common ratio elevated to the power of the n th term). The summation of this series gives you the n th value.

Another common series is the Fibonacci sequence. This is generated by using the sum of the previous term and preceding term before it to create the next number in the sequence. Numbers from this famous series is known as a fibonacci number.