Find the Sum of a Geometric Sequence

A geometric sequence is an ordered set of numbers, or terms in a series, generated via a constant. In a non infinite geometric sequence there is (at least) an identifiable first and last term. This article will help you calculate the sum of any such geometric series.

Steps

1. Calculate the common ratio of the sequence. This is referred to as r. Your strategy to find r depends on how much you already know about the sequence.
   • If you know the first and second term, divide the second term by the first term to find the common ratio.
   • If you know two consecutive terms, divide the latter term by the one the former.
   • If you know the percentage being used to generate the terms, the common ratio is 1 plus the percentage as a decimal. Eg. 4%=1.04, 25%=1.25, 135%=2.35 etc.
2. Identify the first term in the sequence. This is referred to as a. You may be able to easily identify this term, but you may need to use other information you have to solve for the term.
   • If you don't know the first term in the sequence, but you know the common ratio, the last term and the number of terms in the series you can find the first term by solving for a in the following equation:
     
     last term = a (common ratio ^ (number of terms - 1))
     
     or
     
     t_n = ar^{n - 1}
3. Find the number of terms in your sequence. Call this number n. You may be able to just count the terms, but in most cases you'll need to do some work to figure it out.
   • If you do not know the number of terms, but you know the last term, referred to as t_n the first term, and the common ratio you can solve for n using the following equation:
     
     last term = first term (common ratio ^ (n-1))
     
     or
     
     t_n = ar^{n - 1}
4. Solve for the sum. The sum is referred to as S_n. Insert your values into the following equation:
   
   sum of series = first term ( 1 - common ratio ^ number of terms) / (1 - common ratio)
   
   or
   
   S_n=a(1-rⁿ)÷(1-r)

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• Determine Numeric Golden Mean from Geometric Version
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What links here

• Determine Convergence of Infinite Series
• Find Any Term of a Geometric Sequence
• Find the Sum of an Arithmetic Sequence
• Solve Recurrence Relations