Sunday, 14 April 2019

discrete mathematics - How to get kk+1+kk to equate (k+1)k+1?



This is a problem from Discrete Mathematics and its Applications





  1. Let P(n) be the statement that $n!



(a) What is the statement P(2)?
(b) Show that P(2) is true, completing the basis step of the proof.
(c) What is the inductive hypothesis?
(d) What do you need to prove in the inductive step?
(e) Complete the inductive step.
(f) Explain why these steps show that this inequality is true whenever n is an integer greater than 1.




I am currently on part e, completing the inductive step.
Here is my work so far,



I was able to show that the basic step, P(2) is true because 2!<22 or 2<4
Now I am trying to show the inductive step, or P(k)P(k+1)
Assuming P(k), $k! To get (k+1)! on both sides, I multiplied both sides by k+1 to get
(k+1)!<kk(k+1) or (k+1)!<kk+1+kk



How can I get this expression, kk+1+kk to equate (k+1)k+1?



Answer



(n+1)!=(n+1)n!<(n+1)nn<(n+1)(n+1)n=(n+1)n+1


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