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NDA 1 2025 Maths, question 2

Topic: Algebra

Q2

The sum of the first 8 terms of a GP is five times the sum of its first 4 terms. If r≠1r \neq 1 is the common ratio, then what is the number of possible real values of rr?

  1. (a) One
  2. (b) Two
  3. (c) ThreeCorrect answer
  4. (d) More than three

Solution.

Answer: (c) Three

  1. Let the first term be a≠0a \neq 0. The next four are r4r^{4} times the first four, so S8=(1+r4)S4S_{8} = \left( 1 + r^{4} \right)S_{4}.
  2. Hence (r4−4)S4=0\left( r^{4} - 4 \right)S_{4} = 0. If S4≠0S_{4} \neq 0, then r4=4r^{4} = 4, giving r=2r = \sqrt{2} or r=−2r = - \sqrt{2}.
  3. If S4=0S_{4} = 0, then a(1+r)(1+r2)=0a(1 + r)\left( 1 + r^{2} \right) = 0, giving the additional real ratio r=−1r = - 1.
  4. All three satisfy the original equality.
  5. Do not divide by S4S_{4} and lose r=−1r = - 1.