Question

Define the sequence a₁, a₂, a₃ ... as follows:
a₁ = 2, a_n = 5 a_n–1, for all natural numbers n ≥ 2.

Use the Principle of Mathematical Induction to show that the terms of the sequence satisfy the formula a_n = 2.5^n–1 for all natural numbers.

Answer 1

Let P(n) be the statement.

i.e., P(n): a_n = 2.5^n–1 for all natural numbers,

We observe that P(1) is true,

Assume that P(n) is true for some natural number k

i.e., P(k): ak = 2.5^{k – 1}.

Now to prove that P(k + 1) is true.

We have P(k + 1) : a_k+1

= 5.a_k

= 5.(2.5^{k – 1})

= 2.5^k

= `2.5^((k + 1) - 1)`

Thus P(k + 1) is true whenever P(k) is true.

Hence, by the Principle of Mathematical Induction, P(n) is true for all natural numbers.