If the Co-efficients of three successive terms in the expansion of (1 +x)ⁿ are in the ratio 1 : 3 : 5, then find the value of n

If (p+1) ^th  term of an A.P is twice the (q+1)^th terms prove that the (3p+1)^th term is twice the  (p+q+1)^th term

If S _n denotes that Sum of n terms of a G. P., prove that (s₁₀-s₂₀ )² = s₁₀ (s₃₀ - s₂₀)

Show that the coefficient of the middle term in the expansion of (1+x)²ⁿ is equal to the sum of the coefficients of the two middle terms in the expansion of (1+x)^2n-1.

If S₁, S₂, S₃ be respectively the sums of n, 2n, 3n, terms of a G.P. , then prove that S₁ (S₃ - S₂) = (S₂ - S₁)².