Problem 3: Proof by Induction Prove each of the...

Problem 3: Proof by Induction Prove each of the following. (a) $2^n leq (n+1)!$ (b) Let $x_1 = 1$, $x_2 = 2$, and $x_{n+2} = .5 * (x_{n+1} + x_n)$ $forall n in mathbb{N}$. Use Principle of Strong Induction to prove that $1 leq x_n leq 2$ $forall n in mathbb{N}$

Transcript

0:00 Hello everyone.

00:01 So let us prove the following using the proof of induction.

00:06 So here we are having 2 power n less than or equal to n plus 1 the whole factorial for all n greater than or equal to 2.

00:15 So this is the first part of the question.

00:18 First let us take n is equal to 2 so that we get 2 power 2 less than or equal to 3 factorial that is 4 less than or equal to 6.

00:28 So this is true for n is equal to 2 and let us prove for n is equal to k.

00:36 Now we can take 2 power k less than or equal to k plus 1 factorial.

00:47 So here it will be so this can be further written 2 power k less than or equal to here we can write this value as k plus 1 into k factorial.

01:08 So therefore 2 power k is less than or equal to k factorial.

01:13 So which is true for n is equal to k.

01:17 Now let us prove for n is equal to k plus 1 so that we get 2 power k plus 1 less than or equal to k plus 2 factorial.

01:30 So we can write this in terms of 2k dot 2 less than or equal to so this value can be written as 2 k plus 2 into k plus 1 factorial.

01:44 So anyway we already proved that 2k is less than or equal to k plus 1 factorial.

01:50 So we can directly say that 2 power so 2 power k dot 2 which will be less than or equal to k plus 2 factorial.

02:10 So it is true for n is equal to k plus 1.

02:16 So therefore we can say 2 power n less than or equal to n plus 1 the whole factorial is true for all n greater than or equal to 2.

02:34 So this is the answer for the first part of the question.

02:36 Now let us prove the second part of the question.

02:41 Here we are given with certain things x1 and x2 and we have to say that 1 less than or equal to xn less than or equal to 2.

02:54 So this is the part we have to prove and we are given with xn plus 2 which is equal to 0 .5 into xn plus 1 plus xn.

03:03 Now let us take n is equal to 1...

Question

Problem 3: Proof by Induction Prove each of the following. (a) $2^n leq (n+1)!$ (b) Let $x_1 = 1$, $x_2 = 2$, and $x_{n+2} = .5 * (x_{n+1} + x_n)$ $forall n in mathbb{N}$. Use Principle of Strong Induction to prove that $1 leq x_n leq 2$ $forall n in mathbb{N}$

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