Hi there. In this math post, I go over some examples of solving problems involving geometric sequences.
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A geometric sequence of numbers is an ordered list of numbers. This ordered list of numbers is in such a way where the next number is a multiple of the previous number. All the numbers in this ordered list share a common multiple called a common ratio.
Examples of geometric sequences include:
2, 4, 8, 16, 32, ... where the common ratio is 2 as the next term is double of the previous term.
1, 0.1, 0.01, 0.001, ... where the common ratio is 0.1 or one-tenth.
This sequence of -1, +1, -1, +1, -1, +1 includes alternating signs -1, +1, -1, +1, ... where the common ratio is -1.
Geometric sequences can have a limited number of numbers (terminating sequence) or can never ending resulting in an infinite number of terms. (If you want to get more technical, the sequence diverges.)
Given a geometric sequence of numbers, you have the following pieces for finding the n-term in the geometric sequence
To find the n-th term in a geometric sequence, you can use the formula:
Example One
Given a simple geometric sequence of 10, 20, 40, 80, 160, ... what is the twentieth term of the sequence?
The first term is 10 and the common ratio is 2 (from 20 divided by 10, or 160 divided by 80). Since the question is asking for the twentieth term the variable n is 20.
Example Two
What is the tenth term in the geometric sequence of 1, 0.5, 0.25, 0.125, 0.0625, ... ?
The first term is 1 and the common ratio is 0.5 or half (from 0.5 divided by 1). Finding the tenth term involves using n = 10. It may be easier to use the fraction version of the decimal numbers in the sequence.
A geometric sequence has five terms where the common ratio is 3 and the eighth term is 324. What is the first term?
The first term of this particular geometric sequence is 4.
A geometric sequence with a common ratio of 2 starts with 0.5 and ends with 1024, how many terms are there in this geometric sequence?
Start with the geometric sequence formula:
Substitute the known values into their respective variables. The unknown variable here is n in which we are trying to solve for.
Note that 2048 is two to the power of eleven.
Matching the exponents gives 11 = n - 1 which results in the number of terms (n) being 12.
Alternatively, you can use logarithms to solve for n:
Example One
The first term of a geometric sequence is 3 and the eighth term of this sequence is 384. What is the common ratio here?
With the geometric sequence formula, the unknown here is the ratio r.
Example Two - Money Example
Suppose you invest $1000 into a savings account with an interest rate compounded annually. After five years, this initial $1000 compounds into $1610.51 after five years. What is the interest rate?
In the finance context, the formula here is similar to the geometric formula. We have:
where:
(In general, t is the number of compound periods but that is beyond the scope for here.)
Substituting the known values and solving for the interest rate r gives:
This particular example assumes the reader knows how to solving system of equations and is comfortable with (high-school) algebra.
Given a geometric sequence of three numbers which consists of -8, the square root of x and -72, what is the value of x and what is the value of the second term?
With
Substituting values gives these two equations:
Solving for r in the second equation gives:
Substituting this r value into the first equation gives:
The value for x is 576 which means that the second term is the square root of 576 which turns out to be 24. This geometric sequence turns out to be -8, +24, -72. One neat thing about the problem is that we did not need to find the ratio to solve this question. (The common ratio here is -3.)
Thank you for reading.
