If the constant term, in binomial expansion of ( 2xʳ + 1/x² )¹⁰ is 180, then r is equal to __________________.

Solution

Maths sir · 4 steps

Write the general term, let the coefficient 180 fix the term number, then let the zero power fix r.

1. General term. The (k+1)th term is C(10, k)(2xʳ)¹⁰⁻ᵏ(x⁻²)ᵏ. Collect the number and the power of x separately.

   Tₖ₊₁ = C(10, k) 2¹⁰⁻ᵏ xʳ⁽¹⁰⁻ᵏ⁾⁻²ᵏ

2. Use the coefficient. The constant term has coefficient C(10, k)2¹⁰⁻ᵏ = 180. Try k near the end: k=9 gives 10 × 2 = 20, k=8 gives 45 × 4 = 180, k=7 gives 120 × 8 = 960. So k = 8, and no other k works.
3. Use the zero power. For a constant term the power of x is zero: r(10-k) = 2k. With k = 8 this is 2r = 16.
4. So. r = 8.

The idea to keep. When the unknown sits in the power, do not start with the power. Let the given coefficient tell you which term it is, then set the power of x to zero.

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