The sequence u₁, u₂, u₃, ... is such that u₁ = 4 and u_n+1 = 3u_n - 2 for n ≥ 1.

It is given that \( f(n) = 3^{3n} + 6^{n-1} \).

Prove, by mathematical induction, that, for integers n ≥ 2, 4ⁿ > 2n + 3ⁿ.

Prove by mathematical induction that 5^(2n) – 1 is divisible by 8 for every positive integer n.

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…

The curve C has parametric equations x = 4t + 2t³, y = 4t - 2t², for 0 ≤ t ≤ 4. Find the arc length of C, giving your…

The matrix A is given by A = [[2, 2, -3], [2, 2, 3], [-3, 3, 3]] The matrix A has an eigenvector [[1], [-1], [1]]. Fi…