A geometric sequence is a list of terms where each one is the previous term multiplied by a fixed number; a geometric series is what you get when you add those terms up. That one-line distinction decides everything that follows: "find the fifth term" asks for a sequence value, while "find the sum of the first five terms" asks for a series value.

Sequence vs. series at a glance

Geometric sequence Geometric series
What it is The list of terms The sum of the terms
Example 3, 6, 12, 243,\ 6,\ 12,\ 24 3+6+12+243 + 6 + 12 + 24
You compute A single term ana_n A total SnS_n
Formula an=a1rn−1a_n = a_1 r^{n-1} Sn=a1(1−rn)1−rS_n = \dfrac{a_1(1 - r^n)}{1 - r} (r≠1r \ne 1)

In both, a1a_1 is the first term and rr is the common ratio. The defining feature is that ratio: an arithmetic sequence adds the same amount each step, while a geometric sequence multiplies by the same amount each step. If rr is negative the signs alternate; if ∣r∣<1|r| < 1 the terms shrink in size.

When to use which formula

  • Use an=a1rn−1a_n = a_1 r^{n-1} when the question asks for one specific term.
  • Use Sn=a1(1−rn)1−rS_n = \dfrac{a_1(1 - r^n)}{1 - r} when the question asks for a running total and r≠1r \ne 1.
  • If r=1r = 1, every term is identical, so Sn=na1S_n = na_1. The general formula would divide by zero here.
  • For an infinite series, a finite sum exists only when ∣r∣<1|r| < 1; then the terms shrink fast enough for the total to settle.

Worked example: a term and a finite sum

Take the sequence

3, 6, 12, 24, 483,\ 6,\ 12,\ 24,\ 48

so a1=3a_1 = 3 and r=2r = 2.

The fifth term:

a5=3⋅25−1=3⋅16=48a_5 = 3 \cdot 2^{5-1} = 3 \cdot 16 = 48

The sum of the first five terms, added directly:

S5=3+6+12+24+48=93S_5 = 3 + 6 + 12 + 24 + 48 = 93

And the same sum from the formula, as a check:

S5=3(1−25)1−2=3(−31)−1=93S_5 = \frac{3(1-2^5)}{1-2} = \frac{3(-31)}{-1} = 93

Both routes agree, which is exactly the consistency check you want.

Practice it yourself

Take a sequence with first term 55 and common ratio 12\tfrac{1}{2}. Find the first four terms, then their sum; the sum formula should match adding by hand. For a contrast, swap in a negative ratio such as r=−2r = -2 and watch the signs alternate term by term.

Watch out for these slips

  1. Using a common difference (arithmetic thinking) instead of a common ratio.
  2. Mixing up a term question with a sum question.
  3. Applying the finite sum formula when r=1r = 1, which divides by zero.
  4. Forgetting that a negative ratio flips the signs each step.

Geometric patterns show up wherever change happens by a constant factor: doubling, repeated percentage decay, compound growth, and infinite-series ideas in calculus.

Frequently Asked Questions

What is the difference between a geometric sequence and a geometric series?
A geometric sequence is the list of terms, while a geometric series is the sum of those terms. The difference matters because it changes what you compute: find the fifth term asks for a sequence value using the term formula, while find the sum of the first five terms asks for a series value using the sum formula.
How do you find the nth term of a geometric sequence?
Use the formula where the nth term equals the first term times the common ratio raised to the power n minus 1. For the sequence 3, 6, 12, 24, 48 with first term 3 and ratio 2, the fifth term is 3 times 2 to the fourth, which is 3 times 16, giving 48.
What is the formula for the sum of a finite geometric series?
When the common ratio r is not 1, the sum of the first n terms is the first term times the quantity 1 minus r to the n, divided by 1 minus r. If r equals 1, every term is the same and the sum is simply n times the first term, since the usual formula would divide by zero.
When does an infinite geometric series have a finite sum?
An infinite geometric series adds up to a finite value only when the absolute value of the common ratio is less than 1. In that case the terms shrink in size fast enough for the total to settle at a fixed number. If the ratio is 1 or larger in absolute value, the infinite sum does not settle to a finite result.
How is a geometric sequence different from an arithmetic sequence?
In an arithmetic sequence you add the same amount each step, while in a geometric sequence you multiply by the same ratio each step. For example, 3, 6, 12, 24 is geometric because each term doubles. A negative ratio makes the signs alternate, and a ratio with absolute value less than 1 makes the terms shrink in size.

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