The terms are labeled just like they were for an arithmetic sequence: a₁, a₂, a₃, a₄, a₅, etc.

For example, given the sequence, 3, 6, 12, 24, then a₁ = 3, a₂ = 6, a₃ = 12, a₄ = 24,

The numbers in a Geometric Sequence are separated by a common ratio (r),

          Where:      r = a_n / a_n-1

If you look closely at the subscripts, you should be able to state the above equation in simple terms as, "The common ratio is equal to the quotient of any number in the sequence divided by the number that preceded it in the sequence."

In the example given above, r = 6 / 3 = 12 / 6 = 24 / 12 = 2.  It doesn't matter which pair of numbers you use as long as you pick two consecutive numbers in the sequence.

Once the common ratio is known, the next number in the sequence can be determined by multiplying it times the last term.  The formula for this can be easily derived by rearranging the formula for the common difference.

          Where:      a_n = r • a_n-1

Since the last term in the example above was 24 and the common ratio was 2, the next term is 48.

The calculations thus far have been quite simple but what if someone asked you for the 10th term in this sequence?  You could sit down and manually keep multiplying the last term by the common ratio until you got to the 10th term, but that would be very time consuming due to the exponential nature of this series. Note that you will seldom be asked for the 100th term or any similar term as most calculators would give an error due to overflow.

As you might expect, there is a simpler way.  Use the following formula to find the nth term:

          Where:      a_n = a₁ • r ^n-1

All this formula is telling you is that you must multiply the initial number in the sequence by the common ratio raised to the a power which is one less than the term you are looking for.  If you were looking to find the second term you would multiply the first term by the common ratio.  If you were looking to find the third term you would multiply the first term by the common ratio squared.  If you were looking to find the fourth term you would multiply the first term by the common ratio cubed.  Note how you multiply the first term by the common ratio raised to the (n - 1) power.

So if we were looking for the 10th term in the above sequence,

          a_n = a₁ • r ^{n - 1} = 3 • 2⁹ = 3 • 512 = 1536 = a₁₀

What if someone asked you to find the sum of the first 10 terms of the sequence above. Once again, we have yet another formula.

          Where:      S_n = (a₁ - a₁ • rⁿ) / (1 - r)

There really is no simple way to state this formula, so you'll just have to be careful when you make this rather involved computation.  Note how r is raised to the nth power.

In this case, S₁₀ = (3-3•2¹⁰)/(1-2) = (3-3•1024)/(-1) = (3-3072)/(-1) = (-3069)/(-1) = 3069.

Note that the numbers can become very large, very quickly if the common ratio is greater than one. Likewise, they can also become very small, very quickly if the common ratio is less than one. If the common ratio was equal to one, the numbers would not change.

If the common ratio is greater than or equal to one, than the sum of an infinite string of such numbers would be infinite, or undefined.

If the common ratio is less than one then we have a very special case where we can compute the sum of all the terms of such an infinite series, i.e. a geometric series that goes on indefinitely.  The sum of an infinite series where the common ratio is less than one is:

                     S_∞ = a₁ / (1 - r)   if   r < 1

For example, if the series 8, 4, 2, 1 went on indefinitely, the sum of all the terms would be 16 because the common ratio is 0.5 which is less than one.

                     S_∞ = a₁ / (1 - r) = 8/(1-0.5) = 8/0.5 = 16

In closing, note that all geometric sequences have a common ratio.  If they don't, the sequence is not geometric.