If k is the arithmetic mean of two given quantities and p, q are the geometric means between the same two quantities, then p^3 + q^3 is:

Step-by-step solution

Let the two quantities be a and b . Given that k is the arithmetic mean of a and b : k = a + b 2 a + b = 2k Given that p and q are two geometric means between a and b , the sequence a, p, q, b forms a Geometric Progression (G.P.). Let r be the common ratio of this G.P. Then: p = ar q = ar^2 b = ar^3 We need to find the value of p^3 + q^3 : p^3 + q^3 = (ar)^3 + (ar^2)^3 = a^3 r^3 + a^3 r^6 Substituting r^3 = b a : p^3 + q^3 = a^3 ( b a ) + a^3 ( b a )^2 = a^2 b + ab^2 Factoring out ab : p^3 + q^3 = ab(a + b) In a G.