How many terms of the G.P. are needed to give the sum ?

Show that the ratio of the sum of first n terms of a G.P. to the sum of terms from (n + 1)th to (2n)th term is

If a and b are the roots of x² – 3x + p = 0 and c, d are the roots x² – 12x + q = 0, where a, b, c, d form a G.P. Prove that (q + p) : (q – p) = 17 : 15.

A person has 2 parents, 4 grandparents, 8 great grand parents, and so on. Find the number of his ancestors during the ten generations preceding his own.

If S₁, S₂, …., S_n are the sums of n terms of n G.P.’s whose first term is 1 in each and common ratios are 1, 2, 3, …., n respectively, then prove that

S₁ + S₂ + 2S₃ + 3S₄ + … (n – 1) S_n = 1ⁿ + 2ⁿ + 3ⁿ + … + nⁿ.