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The second term of a Geometric Progression (G.P) is 9. If the fourth term is 81, find the common ratio.

Mathematics

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Answer

±3\text{}\pm 3\text{}

Step-by-step solution

3 steps

  1. Write the given information using the formula for the nn-th term of a Geometric Progression, an=arn−1a_n = ar^{n-1}, where aa is the first term and rr is the common ratio.

    The second term is 9:

    a2=ar2−1=ar=9(1)a_2 = ar^{2-1} = ar = 9 \quad (1)

    The fourth term is 81:

    a4=ar4−1=ar3=81(2)a_4 = ar^{4-1} = ar^3 = 81 \quad (2)

  2. Divide equation (2) by equation (1) to eliminate aa and solve for rr.

    ar3ar=819\frac{ar^3}{ar} = \frac{81}{9} r2=9r^2 = 9

  3. Solve for rr.

    r=±9r = \pm\sqrt{9} r=±3r = \pm 3

    The common ratio can be either 3 or −3-3.

    The common ratio is ±3\text{}\pm 3\text{}

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