Matemáticas
The Common Ratio in a Geometric Sequence
Divide a term by the one before it. The same answer every time means a geometric sequence, and that number is also the multiplier for growth or decay.
Common ratio
The fixed number each term of a geometric sequence is multiplied by to give the next one, found by dividing any term by the one before it.
- También llamado
- Multiplier, in percentage growth and decay questions, r
- Dónde lo encuentra el alumnado
- Grade 9 mathematics, and at GCSE and IGCSE in geometric sequences, compound interest and depreciation.
La respuesta corta
The common ratio is the number each term of a geometric sequence is multiplied by to reach the next one. Divide any term by the one before it: in 3, 12, 48, 192 the common ratio is 4. A ratio between 0 and 1 makes the sequence shrink.
Un ejemplo
3, 12, 48, 192 → 12 ÷ 3 = 4, 48 ÷ 12 = 4, 192 ÷ 48 = 4
Every division gives 4, so the common ratio is 4 and the term after 192 is 768. Notice that subtracting instead would have given 9, 36 and 144 — no pattern at all, which is the usual sign that a sequence is geometric rather than arithmetic.
Divide, do not subtract
Take a term and divide it by the one immediately before it. Do it for every consecutive pair you have been given. If the answer is the same each time, the sequence is geometric and that answer is the common ratio.
Two terms are never enough to be sure. Any pair of numbers has a ratio, and any pair also has a difference, so a single division tells you nothing about the sequence as a whole. The claim being made is that the ratio holds everywhere.
When r is a fraction, and when it is negative
A common ratio does not have to be bigger than 1. In 80, 40, 20, 10 each term is half the one before, so the common ratio is 0.5 and the sequence shrinks towards zero without ever reaching it. Anything between 0 and 1 behaves this way.
A negative ratio flips the sign at every step. In 3, −6, 12, −24 the common ratio is −2: the terms alternate between positive and negative while their size doubles. If a sequence swings between positive and negative, a negative ratio is the first thing to test.
Reading a percentage straight off r
This is where the idea earns its keep. A common ratio of 1.05 means each term is 5% more than the last, so savings growing at 5% a year form a geometric sequence. A ratio of 0.92 means each term is 8% less, which is how depreciation questions are set up.
Because the ratio is applied once per step, n steps of growth multiply the starting amount by r raised to the power n. That single fact turns a compound interest question into one line of arithmetic rather than a year-by-year table.
The translation runs both ways. Subtract 1 from the ratio and read what is left as a percentage: 1.03 is a rise of 3%, and 0.75 is a fall of 25%, because it falls 0.25 short of 1.
Preguntas frecuentes
Can the common ratio be 1, or 0?
A ratio of 1 is allowed and gives a constant sequence such as 6, 6, 6, 6. A ratio of 0 is not useful: every term after the first would be zero, and you could no longer divide by a term to recover the ratio. Textbooks exclude 0 for that reason.
How do I tell an arithmetic sequence from a geometric one?
Run both tests on the same terms. Subtract consecutive terms; if the answers match, it is arithmetic. Divide consecutive terms; if those answers match, it is geometric. In 4, 12, 36 the differences are 8 and 24, but the divisions both give 3, so it is geometric with a common ratio of 3.
How do I find r from two terms that are not consecutive?
Divide the later term by the earlier one and take the root matching the number of steps. If the 3rd term is 12 and the 5th is 108, two steps apart, then r² = 108 ÷ 12 = 9, so r = 3 or r = −3. Both fit unless the question says the terms are positive.
Does a geometric sequence with a fractional ratio ever reach zero?
No. Halving 80 gives 40, then 20, then 10, then 5, and each term is half of something positive, so it is still positive. The terms get as close to zero as you like without arriving, which is exactly the behaviour of an exponential decay curve.
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