Question

Prove by method of induction, for all n ∈ N:

Given that t_n+1 = 5t_n + 4, t₁ = 4, prove that t_n = 5ⁿ − 1

Answer 1

The statement P(n) has L.H.S. a recurrence relation t_n+1 = 5t_n + 4, t₁ = 4 and R.H.S. a general statement t_n = 5ⁿ – 1

Step I:

To prove P(1) is true.

L.H.S. = 4, R.H.S. = 5¹ – 1 = 4

∴ P(1) is true.

For n = 2, L.H.S. = t₂ = 5t₁ + 4 = 24

R.H.S. = t₂ = 5² – 1 = 24

∴ P(2) is also true.

Step II:

Assume P(k) is true.

i.e. t_k+1 = 5t_k + 4 and t_k = 5^k – 1

Step III:

To prove P(k + 1) is true.

i.e. to prove t_k+1 = 5^k+1 – 1

Since t_k+1 = 5t_k + 4 and t_k = 5^k – 1 .........(From Step II)

∴ t_k+1 = 5(5^k – 1) + 4 = 5^k+1 – 1

∴ P(k + 1) is true.

From all the steps above,

P(n) : t_n = 5ⁿ – 1 is true for all

n ∈ N where t_n+1 = 5t_n + 4 , t₁ = 4