List down the principle of mathematical induction.

To prove that P(n) is true for all positive integers n, where P(n) is a propositional function

11)   verify that P(1) is true.

22)   Show that the conditional statement P(k) → P(k + 1) is true for all positive 

integers k. The technique that we’ve used in the above example to determine if all the values are true or not, is called mathematical induction. 

Template proof by mathematical induction

1.express to be proved in the form “for all n ≥ b, P(n) for integer b


2.write “Basic Step”. 


Show that P(b) is true,take the correct value of b used is used.


3.write word “Inductive Step”


4.state clearly the inductive hypothesis in the “ assume that p(k) is true for an arbitrary


fixed integer k ≥ b ”


5.proved under assumption that inductive is true


Write P(k + 1) says


6.prove P(k + 1),make assumption P(k)
,

Make sure proof is valid for all integer k with k ≥ b

,
taking care proof work for small value of k, including k = b


7.conclusion of the induction step, saying “ this complete the inductive step”


8.namely that by mathematical induction P(n) is true for all integer n ≥ b

Use of Sequence in Computer Programming

Example:

First of all, we need to write a  program. 

The output should looks like this.

The output of  product is presented in sequence form.