Geometric Sequence Calculator

A geometric sequence is a series of numbers where each element is the previous element multiplied by a constant factor. This factor is called the common ratio. Starting with 2 and repeating this three times, we get 2, 6, 18, 54 etc., which is a sequence with common ratio 3.

This calculator will determine any term of a geometric progression as well as the sum of the first terms and the infinite sum when they exist. Enter either the common ratio and the first term, or the common ratio and any term, or simply two terms. Complete the remaining information to generate the graphical representation of the sequence.

What is a geometric sequence?

A geometric sequence, also known as a geometric progression, is a sequence of numbers where each element after the first is obtained by multiplying the preceding element by a constant called the common ratio and denoted with r. The common ratio must be non-zero.

We call the first term $a_1$ and the common factor $r$. Then the sequence is $a_1,\; a_1 r,\; a_1 r^2,\; a_1 r^3, \ldots$. Since in each step we multiply by $r$, the ratio between any element and the previous one is always the same. This constant ratio is what makes the sequence geometric, and it's a point to check first. If you divide any term by its predecessor, you should get the same number every time.

This is the multiplicative equivalent of an arithmetic sequence. In an arithmetic sequence a constant number is added at each step. Addition leads to steady linear change while multiplication can lead to either accelerating growth or gradual decay. This is typical for quantities that change by percentages.

The formula for the nth term.

It is also possible to determine far away terms without listing each individual term. With an explicit formula it is possible to directly calculate the term at position n.

The exponent is $n-1$ and not $n$, because the first term is already determined before we start multiplying. In the sequence 2, 6, 18, 54, if $a_1 = 2$ and $r = 3$, then the fifth term is $a_5 = 2 \cdot 3^{4} = 2 \cdot 81 = 162$.

The same rule can be expressed recursively. One starts with the first term and multiplies each subsequent term by a constant factor, the ratio.

The recursive form describes a step-by-step calculation while the explicit form is a shortcut to find far away terms with one calculation. This calculator uses an explicit formula so calculating the hundredth term is no more work than calculating the second term.

The sum of a geometric series.

The sum of the individual terms in a geometric sequence results in a geometric series. Because the sum of the first n terms has a simple closed form, there is no need to add them individually.

The principle is very simple. If you form the product of the sum and a certain value and subtract this result from the original sum, almost all terms cancel out, leaving behind a certain value. This is the case for a sequence with five members consisting of 2, 6, 18, 54, and 162: $S_5 = 2 \cdot \frac{1 - 3^5}{1 - 3} = 2 \cdot \frac{-242}{-2} = 242$.

The only exception to this formula is for $r = 1$. In that case the denominator becomes zero. Then all terms are equal to the first term, so the sum is $S_n = n \cdot a_1$. This special handling is taken care of by a calculation tool.

Infinite sums and convergence

It may seem obvious that adding an infinite number of numbers always leads to infinity, but a geometric series can sometimes converge to a finite sum. This is possible when each term goes to zero quickly enough, which happens if the absolute value of the common factor is small enough.

The condition $|r| < 1$ means that $r$ is strictly between $-1$ and $1$. If you are given $a_1 = 6$ and $r = \tfrac{1}{3}$, the sum is $S_\infty = \frac{6}{1 - 1/3} = \frac{6}{2/3} = 9$. Each successive term 6, 2, 2/3 etc. adds up to get closer to 9 but never exceeds it.

If the absolute value of the common factor is greater than 1, then each term either stays the same or increases, so the sum never stabilizes and there is no infinite sum. The calculator will tell you what state your series is in, and only shows an infinite sum if the series converges.

A typical example is Zeno's paradoxes. To cross a room one must first traverse half the distance, then half of what remains and so on, infinitely many times. These distances form a geometric series with common factor one-half, whose sum is equal to the total length of the room. For that reason it is possible in finite time to reach the opposite wall eventually.

Calculating a common factor of two terms

If you know two terms and their respective positions but not the common factor, it can be found by dividing the terms and taking the appropriate root of the quotient.

Suppose the second term is 6 and the fourth term is 54. If $r^{2} = 54/6 = 9$, then $r = 3$. Since the difference of positions is even and the ratio of the two terms is positive, a negative root can also be used, producing $r = -3$ as another sequence that passes through these two terms. The calculator will indicate whether such a second solution exists. If the difference of positions is even but the signs of the two terms are different, then there is no real common factor at all. This is because you cannot turn a positive number into a negative one by raising it to an even power.

Types of geometric series.

The properties of an entire sequence are determined solely by the constant ratio. If you see the value of r, then before calculating each individual term, you can get a rough idea of what the situation is like.

Only two types, the diminishing one and the shrinking alternating one, have an absolute value of constant ratio smaller than one, thus having an infinite sum. In all other cases meaningful numerical values keep accumulating forever.

Applications of geometric series:

Any quantity that changes by a constant factor is in geometric relationship. The compound interest principle describes how capital grows at a fixed rate, and multiplies by the same factor each period of time. Populations doubling from generation to generation, rumors tripling their reach every day, or light beams losing a certain percentage of intensity with each glass pane they pass through - all follow the same rules.

Decay processes are also often observed. Radioactive substances lose a constant proportion of their mass at regular time intervals, a bouncing ball reaches a certain percentage of its previous height on impact, and drugs are excreted from the body in a constant ratio hourly. In all these cases, the value of this factor is known, and one wants to determine the subsequent terms or the total amount - this is exactly what this tool calculates.

Hints for using the calculation tool:

Please enter the position n as an integer. Each term is at position 1, 2, 3 etc. In the constant ratio and terms fields you can directly type in fractions like 1/2 or -3/4 since they will be interpreted as numbers. The result will be shown as a decimal so a constant ratio of one third would show up as 0.3333.

If only two known terms are given, use the "Two Terms" mode to ask the calculator to find the common ratio and first term. To check your answer, look at the table of terms below. Each row is found by multiplying the previous row by the common ratio, and the partial sums column should add up to the total value entered.