Let x_k be the arithmetic mean of the cubes of the first k natural numbers, and y_k be the arithmetic mean of the first k natural numbers. The value of _ k=1 ¹⁰ x_k y_k is:

Step-by-step solution

The arithmetic mean of the cubes of the first k natural numbers is: x_k = 1^3 + 2^3 + + k^3 k = 1 k ( k(k+1) 2 )^2 = k(k+1)^2 4 The arithmetic mean of the first k natural numbers is: y_k = 1 + 2 + + k k = 1 k ( k(k+1) 2 ) = k+1 2 The ratio of x_k to y_k is: x_k y_k = k(k+1)^2 4 k+1 2 = k(k+1) 2 The required sum is: _ k=1 ¹⁰ x_k y_k = _ k=1 ¹⁰ k^2 + k 2 = 1 2 ( _ k=1 ¹⁰ k^2 + _ k=1 ¹⁰ k ) Using the standard summation formulas: _ k=1 ¹⁰ k^2 = 10 11 21 6 = 385 _ k=1 ¹⁰ k = 10 11 2 = 55 Substituting these values into t