A sequence a 1 , a 2 , a 3 , …... is defined by letting a 1 = 3 and a k = 7 a k – 1 for all natural numbers k ≥ 2. Show that a n = 3.7 n-1 for all n ϵ N

Question

Philoid · Accepted Answer

Let P(n): a_n=3.7^n-1 for all n ϵ N

Step1: For n=1,

a₁=3.7^1-1=3

So, it is true for n=1

Step2: For n=k,

Let P(k) be true.

So, a_k=3.7^k-1

Now, we need to show P(k+1) is true whenever P(k) is true.

P(k+1):

a_k+1=7.a_k

=7.3.7^k-1

=3.7^k-1+1

=3.7^(k+1)-1

So, it is true for n=k+1

Hence, by the principle of mathematical induction P(n) is true.

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

A sequence a₁, a₂, a₃, …... is defined by letting a₁ = 3 and a_k = 7 a_{k – 1} for all natural numbers k ≥ 2. Show that a_n = 3.7 ^n-1 for all n ϵ N