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The first three terms of a geometric series are \newline(p−1)(p-1), 22 and \newline(2p+5)(2p+5) respectively, where \newlinepp is a constant.\newlineFind the two possible values of \newlinepp.

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Q. The first three terms of a geometric series are \newline(p−1)(p-1), 22 and \newline(2p+5)(2p+5) respectively, where \newlinepp is a constant.\newlineFind the two possible values of \newlinepp.
  1. Identify common ratio: Identify the common ratio of the geometric series by using the first two terms. The common ratio rr can be calculated by dividing the second term by the first term.\newlineCalculation: r=2(p−1)r = \frac{2}{(p-1)}
  2. Set up equation: Use the common ratio to set up an equation with the second and third terms. The third term should equal the second term multiplied by the common ratio.\newlineCalculation: 2×r=2p+52 \times r = 2p + 5\newlineSubstitute rr from the first step: 2×(2p−1)=2p+52 \times \left(\frac{2}{p-1}\right) = 2p + 5
  3. Simplify and solve: Simplify and solve the equation for pp.\newlineCalculation: 4(p−1)=2p+5\frac{4}{(p-1)} = 2p + 5\newlineCross multiply to clear the fraction: 4=(2p+5)(p−1)4 = (2p + 5)(p-1)\newlineExpand and simplify: 4=2p2+3p−54 = 2p^2 + 3p - 5\newlineRearrange into standard quadratic form: 2p2+3p−9=02p^2 + 3p - 9 = 0

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