Question 47 of 51advanced🔧 ApplyNumerical5 marks

The sum of the first three terms of a GP is 1312\frac{13}{12} and their product is −1. Find the common ratio and the terms.

Correct Answer

The common ratio is -4/3 or -3/4, and the terms are 3/4, -1, 4/3 or 4/3, -1, 3/4.

Exercise: END-OF-CHAPTER EXERCISES | Q: 11 | (Page 22)
For More Understanding

Explanation

The textbook defines a GP as a, ar, ar2ar^{2}... Using a/r, a, ar for three terms simplifies the product to a3a^{3}. Given the product is -1, a=-1. The sum equation yields a quadratic in r, giving two valid common ratios and corresponding terms.

Solution Steps

  1. Step 1: Let the three terms of the GP be a/r, a, ar. Their product is (a/r) * a * ar = a3a^{3} = -1, so a = -1.

  2. Step 2: The sum of the terms is a(1/r + 1 + r) = 13/12. Substituting a = -1 gives -1(1/r + 1 + r) = 13/12, which means 1/r + 1 + r = -13/12.

  3. Step 3: Simplify the sum equation to r + 1/r = -25/12. Multiply by 12r to get the quadratic equation 12r212r^{2} + 25r + 12 = 0.

  4. Step 4: Factorize the quadratic equation as (3r + 4)(4r + 3) = 0, yielding the common ratio r = -4/3 or r = -3/4.

  5. Step 5: If r = -4/3, the terms are 3/4, -1, 4/3. If r = -3/4, the terms are 4/3, -1, 3/4.

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