Linked Questions

3
votes
7answers
324 views

Hint in Proving that $n^2\le n!$ [duplicate]

The problem is to find the values of n such $n^2\le n!$. I have found this set to be {${n\in Z^+| n\le1 \lor n\ge 4}$} I've used a proof by cases to prove the first part ($\forall n\le 1$) which was ...
1
vote
2answers
324 views

Prove by induction that $n! > n^2$ [duplicate]

How does one prove by induction that $n! > n^2$ for $n \geq 4$
3
votes
2answers
253 views

Proof by induction that $n^2 < n!$ [duplicate]

Prove $n^2 < n!$. This is what I have gotten so far basis step: $p(4)$ is true Inductive Hypothesis assume $p(k)$ true for $k \ge 4$ Inductive Step $p(k+1)$ : $(k+1)^2 < (k+1)!$ $$(k+1)^2 =k^...
2
votes
2answers
153 views

Mathematical induction with an inequality and factorial notation: $n! > n^2$ [duplicate]

I'm having difficulty proving $n! > n^2$ for $n \ge 4$ I have previously solved a similar problem but it is $n! > 2^n$. Now I don't know how to solve this. I have only come as far as solving for ...
1
vote
2answers
128 views

Proving $n! > n^2$ by mathematical induction [duplicate]

I'm trying to prove that $n! > n^2$ for $n\geq 4$ by use of mathematical induction, but I get to the inductive step and get lost. But I'm struggling with the inductive step as expected.
1
vote
3answers
83 views

Proof by induction: $n! > n^2$ for $n\ge 4$ [duplicate]

Proof by induction: $n! > n^2$ for $n\ge 4$ Basis step: if $n=4$ $4\cdot3\cdot2\cdot1 > 4^2$ $24 > 16$ I don't know how to do the inductive step.
3
votes
4answers
20k views

Prove that $n!>n^2$ for all integers $n \geq 4$.

I am working on induction problems to prep for Real Analysis for the fall semester. I wanted proof verification and editing suggestions for part (a), and assistance understanding part (b). For part (b)...
4
votes
5answers
279 views

Prove $n!>n^2$ for $n>3$

I'm aware that induction is necessary. I have been stuck on this problem for a few days now. I'm having a hard time understanding how to apply the inductive hypothesis to the inequality to arrive at ...
1
vote
1answer
2k views

Inequalities - proof by induction that $n^2 \leq n!$ for $n\geq 4$

Proof by induction involving inequalities completely escapes me. I've encountered the following problem: For which non-negative integers n is $n^2 ≤ n!$? Prove your answer (by induction). So, ...
4
votes
3answers
204 views

Math induction ($n^2 \leq n!$) help please

I'm having trouble with a math induction problem. I've been doing other proofs (summations of the integers etc) but I just can't seem to get my head around this. Q. Prove using induction that $n^...
2
votes
1answer
321 views

Prove $n! \geq n^2$ for $n \geq 4$

I am working through a discrete math course, and have come upon a question that I don't understand how the solution was obtained. The question is, prove $n! \geq n^2$ Hypothesis: $p(n): n! \geq n^2, ...
1
vote
2answers
270 views

Prove by induction $n! > n^2$

I am trying to prove the inequality in the title for $n\geq 4$; however, I am stuck on the induction step! Any help would be appreciated. For $n\ge 4$, prove that $n! > n^2$. Base Case: $n=4$, ...
1
vote
5answers
116 views

Showing that $n! > n^2$ for $n\geq4$ by induction

My attempt: Prove $ n! > n^2 $ for $ n \geq 4 $ Base Case: $P(4) = 24 > 16$ Inductive Hypothesis $P(k) : k! > k^2 $ $P(k+1) : (k+1)! > (k+1)^2 $ $ (k + 1)! - (k+1)^2 > 0 $ $ (k+...
5
votes
2answers
105 views

Proof using induction: $n! > n^2$, for $n\geq4$

Proof using induction: $n! > n^2$, for $n\geq4$ Basis: For n = 4, we have: $4! > 4^2$ $24 > 16$ (TRUE) Inductive step: By the induction hypothesis: $k! > k^2$ $(k+1)k! > (k+1)k^2$ $(k+...
2
votes
2answers
87 views

Demonstration that $\forall\; n>3,\;\;n^2<n!$

How do I prove by mathematical induction that$$\forall\; n>3,\;\;n^2<n!$$ I tried, $n=4$ then $4^2<4!$ what is true, because $16<24$.$$$$Hypotesis: $n^2<n!$ $$$$Thesis: $(n+1)^2<(n+...

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