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
Let P(n): an=3.7n-1 for all n ϵ N
Step1: For n=1,
a1=3.71-1=3
So, it is true for n=1
Step2: For n=k,
Let P(k) be true.
So, ak=3.7k-1
Now, we need to show P(k+1) is true whenever P(k) is true.
P(k+1):
ak+1=7.ak
=7.3.7k-1
=3.7k-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.