Multiple choice

If the rth, (r + 1)th and (r + 2)th coefficients of (1 + x)n are in AP, then n is a root of the equation

  1. x2 - x(4r + 1) + 4r2 - 2 = 0

  2. x2 + x(4r + 10) + 4r2 - 2 = 0

  3. x2 + x(4r + 1) + 4r2 + 2 = 0

  4. None of these
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A Correct answer
Explanation

If coefficients are in AP, then 2 * C(n, r) = C(n, r-1) + C(n, r+1). Expanding these binomial coefficients and simplifying leads to the quadratic equation in n: n^2 - n(4r + 1) + 4r^2 - 2 = 0.

AI explanation

The coefficients are given by the combinations formula nCr, so the terms nCr-1, nCr, and nCr+1 are in arithmetic progression. Using the property of consecutive binomial coefficients, the middle coefficient nCr equals the average of the other two: nCr = (nCr-1 + nCr+1) / 2. Expanding these combinations and simplifying the resulting equation yields n^2 - n(4r + 1) + 4r^2 - 2 = 0, showing that n is a root of this quadratic equation.

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