Finite geometric series is one of 4 sequences formulas in the algebra section of this library, and it is used at high school · ap level.
Why finite geometric series works
Multiply the whole sum by r and line it up under the original. Every middle term appears in both rows, so subtracting wipes them all out and leaves only the first term and one leftover power. Dividing by 1 minus r finishes the job.
What each symbol means
$a_1$ is the first term, $r$ the common ratio, and $n$ the number of terms.
Finite geometric series: when it holds
$r\ne1$; if $r=1$, then $S_n=na_1$.
When it stops applying
When r equals 1 the formula reads 0 divided by 0 and gives nothing. Every term is then identical, so the sum is simply n times a1: adding five copies of 7 gives 35, which the formula cannot produce.
Finite geometric series: a worked example
$1+2+4+8=1(1-2^4)/(1-2)=15$.
The mistake to avoid
What people do: Using the exponent of the last term instead of the number of terms.
Why it goes wrong: For 1 + 2 + 4 + 8 there are four terms, so n is 4 and the answer is 15. Using 3, because the last term is 2 cubed, gives 7 and quietly drops the largest term.
Do this instead: Count the terms you are actually adding, including the first one, and use that count as n.
Finite geometric series: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $a_1$ is the first term, $r$ the common ratio, and $n$ the number of terms.
- Check the conditions before substituting. $r\ne1$; if $r=1$, then $S_n=na_1$.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most algebra slips.
Where this formula fits
- Subject
- Algebra formulas — 28 entries in this library
- Topic
- Sequences
- Level
- High school · AP
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where finite geometric series comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Sequences and Series — the lesson behind this formula: recognize arithmetic, geometric, and recursive patterns.
- Algebra Calculator — check your substitution and the value it produces.
- Study algebra — the subject guide that explains the ideas these formulas compress.
- Algebra I Practice — questions that make you retrieve the formula instead of recognising it.
- All 28 algebra formulas — the full grouped reference, or the complete formula library.
Questions about finite geometric series
Does the series have to begin with the first term of the sequence?
No. Treat whichever term you start at as a1 and count from there, since a geometric sequence looks the same from any starting point once you relabel.
What if the ratio is negative?
It works without change. For 1 - 3 + 9 - 27 with r = -3 and n = 4, the formula gives (1 - 81) divided by 4, which is -20, matching the direct addition.
Why do some books write it as a1 times (r to the n minus 1) over (r minus 1)?
Because multiplying top and bottom by -1 gives the same value. That version avoids negative signs when r is bigger than 1, so it is more comfortable for growth problems.
How is this different from the infinite version?
This one adds a fixed number of terms and works for any ratio except 1. The infinite version needs the terms to shrink, so it requires the ratio to sit strictly between -1 and 1.