It is given that y = xe^ax, where a is a constant. Prove by mathematical induction that, for all positive integers n, dⁿy/dxⁿ = (aⁿx + naⁿ⁻¹)e^ax.

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 …

Given that a is a constant, prove by mathematical induction that, for every positive integer n, dⁿ/dxⁿ(xeᵃˣ) = naⁿ⁻¹e…

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

A curve has equation x² – 6xy + 25y² = 16. Show that dy/dx = 0 at the point (3, 1). By finding the value of d²y/dx² a…

Let In = ∫₀^π xⁿ sin x dx, where n is a non-negative integer. Show that In = n(π)ⁿ⁻¹ – n(n − 1)In-2, for n ≥ 2. Find …

By considering ∑_{r=1}^{n} z²ʳ⁻¹, where z = cos θ + i sin θ, show that, if sin θ ≠ 0, ∑_{r=1}^{n} sin (2r − 1)θ = sin…