# proof by mathematical induction

posted by .

subject is PreCalulus.

2^(k+3) = and < (k+3)!

i know how to do proving using mathematical induction when its just an equal sign, but I don't understand what to do when its an inequality.

thank you!!!

Here is what I would do

Step 1: check if it is true for k=1

that is: let k=1
2^4<= 4! ?
16<=24 ? YES

Step 2: assume it is true for k=n

or 2^(n+3) <= (n+3)! is true

Step 3: test to see if it is then true for k= n+1

or, is 2^(n+4) <= (n+4)! ??

subtract the two inequations (step 4 minus step3)

is 2^(n+4) - 2^(n+3) <= (n+4)! - (n+3)! ???

the left side factors to
2^(n+3)(2-1)
or 2^(n+3)

the right side is (n+4)*(n+3)! .... {just like 8! - 7! = 8*7!}

so we are looking at:

is 2^(n+3) <= (n+4)*(n+3)! ??

Well, the left side is the original left side of step 2, and the right side is the original right side of step 2 multiplied by a positive number > 1.

Since the right side was already >= the left side, muliplying it by a positive integer would cerainly make it even larger.

thank you lots!!

3467800

## Similar Questions

1. ### mathematical induction

solve this using the mathematical induction : 1/1(3)+1/3(5)+1/5(7)+......+1/(2n-1)(2n+1)=n/(2n+1)
2. ### AP Calc

Use mathematical induction to prove that the statement holds for all positive integers. Also, can you label the basis, hypothesis, and induction step in each problem. Thanks 1. 2+4+6+...+2n=n^2+n 2. 8+10+12+...+(2n+6)=n^2+7n
3. ### Calculus

Use mathematical induction to prove that the statement holds for all positive integers. Also, label the basis, hypothesis, and induction step. 1 + 5 + 9 + … + (4n -3)= n(2n-1)
4. ### Mathematical induction. I'm stuck. So far I have..

For all integers n ≥ 1, prove the following statement using mathematical induction. 1+2^1 +2^2 +...+2^n = 2^(n+1) −1 Here's what I have so far 1. Prove the base step let n=1 2^1=2^(1+1)-1 False. Someone else suggested that …
5. ### Pre-Calculus

What makes a mathematical induction true?
6. ### precalculus

Can you please check my answers? 1.Find Pk + 1 if Pk=2^K-1/k! answer: 2^k+1/(k+1)! 2.Find Pk + 1 if Pk = 7 + 13 + 19 + ...+[6(k - 1)+1] + (6k + 1) answer: 7+13+9...(6k-1+1)+6k+1 +(6k+2) 3.What is the first step when writing a proof
7. ### precalculus

Find Pk + 1 if Pk = 7 + 13 + 19 + ...+[6(k - 1)+1] + (6k + 1) 7 + 13 + 19 + …+[6(k - 1) + 1] + (6k + 1) + [6(k + 1) + 1] 8 + 14 + 20 + …+[7(k - 1) + 1] + (7k + 1) 7 + 13 + 19 + …+(6k + 1) 7 + 13 + 19 + ...+[6(k - 1) + 1] + (6k7 …
8. ### precalculus

How would you describe a proof by mathematical induction?
9. ### Mathematical Induction

I have been given that a1 = 1 and an+1 = 1/3*(an + 4). In order to prove that this sequence is monotonous, what is the second step of mathematical induction?
10. ### Mathematical Induction

Use mathematical induction to prove that the following is true. 8+11+14...+(3n+5)=1/2n(3n+13), for all n in the set of natural numbers.

More Similar Questions