It is given that ∑ (from r=1 to n) u_r = n²(2n + 3), where n is a positive integer.

Given that f(r) = 1 / ((r + 1)(r + 2)), show that

Use the method of differences to show that Σ_{r=1}^N 1/((2r + 1)(2r + 3)) = 1/6 - 1/(2(2N + 3)). Deduce that Σ_{r=N+1…

Expand and simplify (r + 1)⁴ – r⁴. Use the method of differences together with the standard results for Σr and Σr² to…

Use de Moivre's theorem to show that cos 5θ = cos θ(16 sin⁴ θ – 12 sin² θ + 1). By considering the equation cos 5θ = …

Let I_n = ∫₀¹ (1-x)ⁿeˣ dx. Show that, for all positive integers n, I_n = nI_(n-1) - 1. Find the exact value of I_4. B…

Given that x(d²y/dx²) + (2x+2)(dy/dx) + (2-3x)y = 10e^(2x) and that v = xy, show that d²v/dx² + 2(dv/dx) – 3v = 10e^(…