The mean of the coefficients of x, x², …, x⁷ in the binomial expansion of (2+x)⁹ is ___________.

Solution

Maths sir · 4 steps

Do not add seven coefficients one by one: put x = 1 for the full sum, then remove the three you do not need.

1. General term. In (2+x)⁹ the coefficient of xʳ is C(9, r)2⁹⁻ʳ. You need the sum of these for r = 1 to 7.
2. Sum of all. Putting x = 1 adds every coefficient from r = 0 to 9: (2+1)⁹ = 3⁹ = 19683.
3. Remove the extras. Subtract the coefficients you do not want. For r = 0 it is 2⁹ = 512, for r = 8 it is C(9, 8) × 2 = 18, and for r = 9 it is 1.

   Σᵣ₌₁⁷C(9, r)2⁹⁻ʳ = 19683 - 512 - 18 - 1 = 19152

4. So. There are 7 coefficients, so the mean is 19152/7 = 2736.

The idea to keep. For the sum of some coefficients, put x = 1 to get the sum of all of them, then subtract the few end terms you do not need. That is much faster than adding term by term.

Answer from the NTA final answer key. Solution by PrepWiser.