Thread: Mathematical inductions

1. Mathematical inductions

Hi

i have a couple of exercises in math and i am not sure if i solved them right. Can someone please tell me what the solutions are?

First principal mathematical induction

1/(1*3)+ 1/(3*5)+...+1/(2n-1)(2n+1)=n/2n+1

Second principle of mathematical induction

n!<n^n for n>1

thank you in advance

2. Originally Posted by tironci
Hi

i have a couple of exercises in math and i am not sure if i solved them right. Can someone please tell me what the solutions are?

First principal mathematical induction

1/(1*3)+ 1/(3*5)+...+1/(2n-1)(2n+1)=n/2n+1
Let $P(n):$ " $\frac 1{1 \cdot 3} + \frac 1{3 \cdot 5} + \cdots + \frac 1{(2n - 1)(2n + 1)} = \frac n{2n + 1}$" for all $n \in \mathbb{N}$ (i take $\mathbb{N}$ to be the positive integers here)

Step 1: Show that $P(1)$ is true.

Step 2: Assume $P(n)$ is true and show that this implies $P(n + 1)$ is true

Then the claim will hold for all $n \in \mathbb{N}$ by mathematical induction

Second principle of mathematical induction

n!<n^n for n>1

thank you in advance
how did you define the second principle of induction. in a class i am taking we did three forms, and i know of another from a previous class. i want to make sure we are on the same page.

3. in class we did only the first principle of induction and the second principle of induction which is also called "Strong Induction"

Thank you for your help