If the 17^th and 18^th terms in the expansion of (2 + a)⁵⁰ are equal, find the value of a.

If the coefficients of 2^nd, 3^rd and 4^th terms in the expansion of (1 + x)²ⁿ are in AP, show that 2n² – 9n + 7 = 0.

Find the 6^th term of the expansion (y^1/2 + x^1/3)ⁿ, if the binomial coefficient of the 3^rd term from the end is 45.

Find the coefficient of x⁴ in the expansion of (1 + x)ⁿ (1 – x)ⁿ. Deduce that C₂ = C₀C₄ – C₁C₃ + C₂C₂ – C₃C₁ + C₄C₀, where C_r stands for ⁿC_r.

Prove that the coefficient of xn in the binomial expansion of (1 + x)²ⁿ is twice the coefficient of xⁿ in the binomial expansion of (1 + x)^2n-1.