Let n ≥ 5 be an integer. If 9ⁿ - 8n - 1 = 64α and 6ⁿ - 5n - 1 = 25β, then α - β is equal to

Solution

Maths sir · 4 steps

Write 9 as 1 + 8 and 6 as 1 + 5, expand both, and the nC2 terms cancel when you subtract.

1. Binomial expansion. Write 9ⁿ = (1+8)ⁿ = 1 + ⁿC₁ 8 + ⁿC₂ 8² + ⁿC₃ 8³ + … + ⁿCₙ 8ⁿ. The first two terms are exactly 1 + 8n, so they cancel with -8n - 1.
2. Find alpha. Every term left has at least 8² = 64 in it. Taking 64 out lowers each power of 8 by two.

   α = ⁿC₂ + ⁿC₃ 8 + ⁿC₄ 8² + … + ⁿCₙ 8ⁿ⁻²

3. Find beta. In the same way, 6ⁿ = (1+5)ⁿ gives 6ⁿ - 5n - 1 = 25(ⁿC₂ + ⁿC₃ 5 + … + ⁿCₙ 5ⁿ⁻²). So β = ⁿC₂ + ⁿC₃ 5 + ⁿC₄ 5² + … + ⁿCₙ 5ⁿ⁻².
4. So. Subtract term by term. The ⁿC₂ terms cancel, leaving α - β = ⁿC₃(8-5) + ⁿC₄(8²-5²) + … + ⁿCₙ(8ⁿ⁻²-5ⁿ⁻²), which is option (c).

The idea to keep. For aⁿ - (a-1)n - 1, write a = 1 + (a-1) and expand. The first two terms cancel and the rest are divisible by (a-1)². Track the powers carefully after you divide.

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