Find The Common Ratio of The Following Geometric Sequence Calculator

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. This calculator helps you find the common ratio between consecutive terms of a geometric sequence.

What is a Common Ratio?

The common ratio in a geometric sequence is the constant factor that is multiplied by each term to obtain the next term in the sequence. For example, in the sequence 2, 6, 18, 54, the common ratio is 3 because each term is multiplied by 3 to get the next term.

Geometric sequences are fundamental in mathematics and appear in various real-world applications, including finance, physics, and biology.

How to Find the Common Ratio

To find the common ratio of a geometric sequence, you need at least two consecutive terms of the sequence. The common ratio (r) can be calculated by dividing the second term by the first term.

r = term₂ / term₁

If the sequence has more than two terms, you can verify the common ratio by dividing any subsequent term by its immediate predecessor. If all these divisions yield the same value, the sequence is geometric.

Formula

The formula to find the common ratio of a geometric sequence is straightforward:

Common Ratio (r) = Second Term (term₂) / First Term (term₁)

This formula works for any two consecutive terms in a geometric sequence. If the sequence is not geometric, the common ratio will not be consistent across all terms.

Worked Example

Let's find the common ratio of the geometric sequence: 5, 15, 45, 135.

Step 1: Identify the first two terms.

First term (term₁) = 5

Second term (term₂) = 15

Step 2: Apply the formula.

Common Ratio (r) = term₂ / term₁ = 15 / 5 = 3

Step 3: Verify with other terms.

term₃ / term₂ = 45 / 15 = 3

term₄ / term₃ = 135 / 45 = 3

Since all ratios are equal, the common ratio is confirmed to be 3.

FAQ

What if the common ratio is negative?

A negative common ratio is valid in geometric sequences. For example, the sequence -2, 4, -8, 16 has a common ratio of -2 because each term is multiplied by -2 to get the next term.

Can the common ratio be a fraction?

Yes, the common ratio can be any real number, including fractions. For example, the sequence 3, 1.5, 0.75 has a common ratio of 0.5.

How do I know if a sequence is geometric?

A sequence is geometric if the ratio between consecutive terms is constant. You can check this by dividing any term by its predecessor and verifying that the result is the same throughout the sequence.