The first term of an A.P. of 30 non-negative terms is 10 3 . If the sum of this A.P. is the cube of its last term, then its common difference is:

Step-by-step solution

Given a = 10 3 and n = 30 . Let the last term be l and the common difference be d . The sum of the A.P. is given by S₃₀ = 30 2 (a + l) = 15 ( 10 3 + l ) . According to the given condition, S₃₀ = l^3 . 15 ( 10 3 + l ) = l^3 50 + 15l = l^3 l^3 - 15l - 50 = 0 (l - 5)(l^2 + 5l + 10) = 0 Since l^2 + 5l + 10 = 0 has no real roots, we get l = 5 . The last term of the A.P. is l = a + 29d . 10 3 + 29d = 5 29d = 5 - 10 3 = 5 3 d = 5 87 Answer: 5 87