The ratio of the A.M. and G.M. of two positive numbers a and b, is m : n. Show that

If a, b, c, d are in G.P, prove that (aⁿ + bⁿ), (bⁿ + cⁿ), (cⁿ + dⁿ) are in G.P.

If a and b are the roots of x² – 3x + p = 0 and c, d are roots of x² – 12x + q = 0, where a, b, c, d form a G.P. Prove that (q + p): (q – p) = 17:16.

If a, b, c are in A.P.; b, c, d are in G.P. and

are in A.P. prove that a, c, e are in G.P.

Find the sum of the following series up to n terms:

6 + 66 + 666 + …