A geometric sequence is a list of numbers in which each term is obtained by multiplying the previous term by a constant called the common ratio. Recognising this pattern helps find later terms and calculate the total of a finite sequence without adding every term individually.

nth-term formula

aₙ = a₁ × rⁿ⁻¹

Here a₁ is the first term, r is the common ratio, and n is the position of the term. To find r, divide a term by the preceding term when that preceding term is not zero.

Example: finding a term

Consider 3, 6, 12, 24, … . The first term is 3 and the common ratio is 2. The fifth term is a₅ = 3 × 2⁴ = 48.

Sum of a finite geometric sequence

When r is not 1, the sum of the first n terms is Sₙ = a₁(1 − rⁿ) ÷ (1 − r). An equivalent form is a₁(rⁿ − 1) ÷ (r − 1). If r = 1, every term equals a₁ and the sum is n × a₁.

For the first four terms 3, 6, 12, 24, the sum is 3 + 6 + 12 + 24 = 45. The formula gives 3(1 − 2⁴) ÷ (1 − 2) = 45.

How to use the formulas

  1. Identify the first term and common ratio.
  2. Confirm that the ratio is consistent across the sequence.
  3. For a specific term, substitute its position into the nth-term formula.
  4. For a finite total, use the sum formula and check the r = 1 case.
  5. Compare the result with the first few terms to catch errors.

Common mistakes

The exponent in the nth-term formula is n − 1, not n, because the first term has not been multiplied by the ratio. A ratio can be negative or fractional; this changes the signs or size of later terms but does not change the formula.

Frequently asked questions

How do I identify a geometric sequence?

Check whether dividing each term by the previous non-zero term gives the same ratio.

Can the common ratio be negative?

Yes. A negative ratio generally alternates the signs of consecutive non-zero terms.

Is the infinite sum formula the same?

No. An infinite geometric series has a finite sum only when the absolute value of the ratio is less than 1.