Let ( 2x² + 3x + 4 )¹⁰ = Σᵣ₌₀²⁰ aᵣxʳ
Then a₇/a₁₃ is equal to ______.

Solution

Maths sir · 3 steps

Do not expand: replace x by 2/x and the polynomial comes back, linking aᵣ with a₂₀₋ᵣ.

1. Spot the symmetry. Let f(x) = 2x² + 3x + 4. The outer coefficients are 2 and 4, so try x → 2/x: f(2/x) = 8/x² + 6/x + 4 = 2/x²(4 + 3x + 2x²) = 2/x² f(x).
2. Raise to the tenth. Raising both sides to the power 10 gives Σ aᵣ 2ʳ/xʳ = 2¹⁰/x²⁰ Σ aᵣ xʳ. On the right, the coefficient of x⁻ʳ comes from the term a₂₀₋ᵣx²⁰⁻ʳ.

   2ʳ aᵣ = 2¹⁰ a₂₀₋ᵣ ⇒ aᵣ/a₂₀₋ᵣ = 2¹⁰⁻ʳ

3. So. Take r = 7, so 20 - r = 13: a₇/a₁₃ = 2¹⁰⁻⁷ = 2³ = 8.

The common slip: 0.125. Writing the relation the wrong way round, as a₁₃/a₇ = 8, gives 0.125 instead of 8.

The idea to keep. When a question asks for a ratio of coefficients aᵣ and aₙ₋ᵣ, look for a substitution like x → k/x that maps the polynomial back onto itself. Then compare coefficients of the same power.

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