A sequence d₁, d₂, d₃ …. Is defined by letting d₁ = 2 and for all natural numbers, k ≥ 2. Show that for all n ϵ N.

A sequence a₁, a₂, a₃ ... is defined by letting a₁ = 3 and a_k = 7a_k–1 for all natural numbers k ≥ 2. Show that an = 3.7^n–1 for all natural numbers.

A sequence b₀, b₁, b₂ ... is defined by letting b₀ = 5 and b_k = 4 + b_{k – 1} for all natural numbers k. Show that b_n = 5 + 4n for all natural number n using mathematical induction.

Prove that for all n ϵ N

Cos α + cos (α + β) + cos (α + 2β) + …. + cos (α + (n – 1)β)

Prove that, cos θ cos 2θ cos2²θ ….. cos 2^n–1θ for all n ϵ N.