It is given that u_r = r × r! for r = 1, 2, 3, ... . Let S_n = u_1 + u_2 + u_3 + ... + u_n. Write down the values of 2! - S_1, 3! - S_2, 4! – S_3, 5! – S_4. Conjecture a formula for S_n. Prove, by mathematical induction, a formula for S_n, for all positive integers n.

It is given that y = (1 + x)² ln(1 + x). Find d³y/dx³. Prove by mathematical induction that, for every integer n ≥ 3,...

It is given that y = ln(ax + 1), where a is a positive constant. Prove by mathematical induction that, for every posi...

Find a matrix A whose eigenvalues are -1, 1, 2 and for which corresponding eigenvectors are (1) (0), (0) (1) (1), (1)...

Using factorials, show that (n/(r-1)) + (n/r) = ((n+1)/r). Hence prove by mathematical induction that (a + x)ⁿ = (n/0...

The linear transformation T : R⁴ → R⁴ is represented by the matrix A, where A = (1 3 5 7) (2 8 7 9) (3 ...