How to Find the Common Ratio of a Fraction

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Calculating the common ratio of a geometric series is a skill you learn in calculus and is used in fields ranging from physics to economics. A geometric series has the form "a*r^k", where "a" is the first term of the series, "r" is the common ratio and "k" is a variable. The terms of the series are frequently fractions. The common ratio is the constant you multiply each term by to generate the next term. You can use the common ratio to calculate the sum of the series.

  • Write down any two sequential terms of the geometric series, preferably the first two. For example, if your series is 3/2 + -3/4 + 3/8 + -3/16 + .. your can use 3/2 and -3/4.

  • Divide the second term by the first term to find the common ratio. To divide fractions, flip the divisor and make it multiplication. Using the previous example with 3/2 and -3/4, the common ratio is (-3/4)/(3/2) = (-3/4)*(2/3) = -6/12 = -1/2.

  • Use the common ratio, the first term and the total number of terms to calculate the sum of the series. If you have a finite number of terms, use the formula "a*(1-r^n)/(1-r)", where "a" is the first term, "r" is the common ratio and "n" is the number of terms. Use the formula "a/(1-r)" if the series is infinite, where "a" is the first term and "r" is the common ratio. The terms must approach 0 for the series to converge and have a sum. Using the previous example, the common ratio is -1/2, the first term is 3/2 and the series is infinite, so the sum is "(3/2)/(1-(-1/2)) = 1."

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