A sequence `b_(0),b_(1),b_(2), . . .` is defined by letting `b_(0)=5` and `b_(k)=4+b_(k-1)`, for all natural number k. Show that `b_(n)=5+4n`, for all natural number n using mathematical induction.

A sequence `b_(0),b_(1),b_(2), . . .` is defined by letting `b_(0)=5`

1.	A sequence `b_(0),b_(1),b_(2), . . .` is defined by letting `b_(0)=5` and `b_(k)=4+b_(k-1)`, for all natural number k. Show that `b_(n)=5+4n`, for all natural number n using mathematical induction.
Answer» Consider the given statement, `P(n):b_(n)=5+4n`, for natural numbers given that `b_(0)=5` and `b_(k)=4+b_(k-1)` Step I P(1) is true `P(1):b_(1)=5+4xx1=9` As `b_(0)=5,b_(1)=4+b_(0)=4+5=0` Hence, P(1) true. Step II Now, assume that P(n) true for n=k. `P(k):b_(k)=5+4k` Step III Now, to prove P(k+1) is true, we have to show that `:. P(k+1):b_(k=1)=5+4(k+1)` `b_(k+1)=4+b_(k+1-1)` `=4+b_(k)` `=4+5+4k=5+4(k+1)` So, by the mathematical induction P(k+1) is true whenever p(k) is ture, hence P(n) is true.