# Math Help - 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

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