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Determine the common ratio of a geometric sequence

The common ratio is the constant multiplicative factor relating each term to the preceding term, found as an+1/ana_{n+1}/a_n when the preceding term is nonzero; it may be an integer, fraction, decimal, or negative number, and the same factor must generate every consecutive pair. This distinguishes multiplicative change from the constant difference of an arithmetic sequence and supports writing an=a1rn−1a_n=a_1r^{n-1} and interpreting geometric growth, decay, and geometric series; complex ratios and advanced degenerate or generalized cases are outside this scope.

Detailed Explanation: Determine the common ratio of a geometric sequence

To determine the common ratio of a geometric sequence, divide each term by the term immediately before it:

r=an+1anr=\frac{a_{n+1}}{a_n}

The ratio must be the same for every pair of consecutive terms. A common ratio can be positive, negative, an integer, a fraction, or a decimal.

Example: Find the common ratio of the sequence

48, −24, 12, −6,…48,\ -24,\ 12,\ -6,\ldots

Step 1: Divide the second term by the first term.

r=−2448=−12r=\frac{-24}{48}=-\frac{1}{2}

Step 2: Check another consecutive pair.

12−24=−12\frac{12}{-24}=-\frac{1}{2}

Check the next pair as well:

−612=−12\frac{-6}{12}=-\frac{1}{2}

Since the same value is produced each time, the common ratio is

r=−12\boxed{r=-\frac{1}{2}}

This means each term is found by multiplying the previous term by −12-\frac{1}{2}. The negative sign causes the terms to alternate signs, and the factor 12\frac{1}{2} makes their sizes decrease.

Learn by doing: Determine the common ratio of a geometric sequence

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