HesapHanem

Common Ratio Calculator

Enter two consecutive terms to find the common ratio of a geometric sequence.

Result

-0.25
5.00

How it is calculated

The tool takes the first two terms a1 and a2 and computes r = a2 ÷ a1. The next term, which is the third term of the sequence, is a3 = a2 × r. Both results are rounded to two decimals for display, and the next term is calculated from the unrounded ratio, so it can differ slightly from the shown ratio times the second term. Terms can be negative or decimals. A negative ratio means the signs alternate. A first term of zero gives no result because the division is impossible. A second term of zero gives r = 0 and a next term of 0, a degenerate case rather than a real geometric sequence. The tool uses only two terms and cannot tell whether the longer sequence really is geometric.

Worked example

The sequence starts 80, −20. Common ratio: −20 ÷ 80 = -0.25. Next term: −20 × −0.25 = 5.00. The tool displays a common ratio of -0.25 and a next term of 5.00. The sequence therefore continues 80, −20, 5, −1.25 and so on, with signs alternating and the size shrinking by a quarter each step.

Try this example →

Common mistakes

  • Subtracting instead of dividing. A common difference of −100 is an arithmetic sequence idea, while a geometric sequence uses the ratio −0.25 for 80 and −20.
  • Dividing in the wrong direction. The ratio is the second term over the first. Entering 20 then 80 gives 4, not 0.25.
  • Expecting the next term to be the second one. The tool gives the term after the two you entered, so 80, −20 returns the third term, 5.00.
  • Ignoring the sign. A negative ratio is valid and produces alternating positive and negative terms. It is not an error.
  • Assuming a sequence is geometric from two terms alone. Any two nonzero numbers produce a ratio, so check a third term yourself before trusting it.

About this calculator

The common ratio of a geometric sequence is the fixed number you multiply each term by to get the next one, found by dividing any term by the term immediately before it: r = a(n+1) ÷ a(n). Unlike an arithmetic sequence, where consecutive terms differ by a constant amount, a geometric sequence grows or shrinks by a constant factor, which is why sequences like 3, 6, 12, 24 (r = 2) or 100, 50, 25, 12.5 (r = 0.5) either compound upward or decay toward zero.

Students and teachers use the common ratio to identify and continue geometric sequences in algebra courses, while it also underpins real-world compounding models — a savings account compounding at a fixed rate, radioactive decay, or exponential population growth are all geometric sequences under the hood, with the common ratio playing the role of the growth or decay factor.

Divide any term of a geometric sequence by the one before it and you have the common ratio; enter two consecutive terms to see it and the term that would come next.

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