Proving $3^n < (n+3)!$ for all natural numbers $n$ [closed]
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I have the problem:
Prove: $3^n < (n+3)!$ for all natural numbers $n$.
I get to the point,
$$P(k+1) = 3^{k} cdot3 < (k+3)! cdot3 < (k+3)! cdot(k+3)$$
but I am not sure where to go from here.
Help appreciated.
discrete-mathematics
edited Nov 27 at 4:40
Blue
47.2k870149
asked Nov 27 at 1:39
cap
83
closed as unclear what you're asking by max_zorn, Shailesh, Lord Shark the Unknown, T. Bongers, user10354138 Nov 27 at 5:22
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
up vote
-2
down vote
favorite
I have the problem:
Prove: $3^n < (n+3)!$ for all natural numbers $n$.
I get to the point,
$$P(k+1) = 3^{k} cdot3 < (k+3)! cdot3 < (k+3)! cdot(k+3)$$
but I am not sure where to go from here.
Help appreciated.
discrete-mathematics
edited Nov 27 at 4:40
Blue
47.2k870149
asked Nov 27 at 1:39
cap
83
closed as unclear what you're asking by max_zorn, Shailesh, Lord Shark the Unknown, T. Bongers, user10354138 Nov 27 at 5:22
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
1
1) If you are going to delete a question, delete it. Don't just edit so it is unreadable. and 2) Don't delete questions. Just because you now know the answer doesn't mean other people will want to use it.
– fleablood
Nov 27 at 3:11
1
Do not vandalize your question to obfuscate it after posting. It's very rude to the person who bothered to answer.
– T. Bongers
Nov 27 at 4:31
What on earth are $P$ and $k$?
– user21820
yesterday
add a comment |
up vote
-2
down vote
favorite
up vote
-2
down vote
favorite
I have the problem:
Prove: $3^n < (n+3)!$ for all natural numbers $n$.
I get to the point,
$$P(k+1) = 3^{k} cdot3 < (k+3)! cdot3 < (k+3)! cdot(k+3)$$
but I am not sure where to go from here.
Help appreciated.
discrete-mathematics
edited Nov 27 at 4:40
Blue
47.2k870149
asked Nov 27 at 1:39
cap
83
I have the problem:
Prove: $3^n < (n+3)!$ for all natural numbers $n$.
I get to the point,
$$P(k+1) = 3^{k} cdot3 < (k+3)! cdot3 < (k+3)! cdot(k+3)$$
but I am not sure where to go from here.
Help appreciated.
discrete-mathematics
discrete-mathematics
edited Nov 27 at 4:40
Blue
47.2k870149
asked Nov 27 at 1:39
cap
83
edited Nov 27 at 4:40
Blue
47.2k870149
asked Nov 27 at 1:39
cap
83
edited Nov 27 at 4:40
Blue
47.2k870149
edited Nov 27 at 4:40
Blue
47.2k870149
edited Nov 27 at 4:40
Blue
47.2k870149
47.2k870149
asked Nov 27 at 1:39
cap
83
asked Nov 27 at 1:39
cap
83
asked Nov 27 at 1:39
cap
83
83
closed as unclear what you're asking by max_zorn, Shailesh, Lord Shark the Unknown, T. Bongers, user10354138 Nov 27 at 5:22
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
closed as unclear what you're asking by max_zorn, Shailesh, Lord Shark the Unknown, T. Bongers, user10354138 Nov 27 at 5:22
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
1
1) If you are going to delete a question, delete it. Don't just edit so it is unreadable. and 2) Don't delete questions. Just because you now know the answer doesn't mean other people will want to use it.
– fleablood
Nov 27 at 3:11
1
Do not vandalize your question to obfuscate it after posting. It's very rude to the person who bothered to answer.
– T. Bongers
Nov 27 at 4:31
What on earth are $P$ and $k$?
– user21820
yesterday
add a comment |
1
1) If you are going to delete a question, delete it. Don't just edit so it is unreadable. and 2) Don't delete questions. Just because you now know the answer doesn't mean other people will want to use it.
– fleablood
Nov 27 at 3:11
1
Do not vandalize your question to obfuscate it after posting. It's very rude to the person who bothered to answer.
– T. Bongers
Nov 27 at 4:31
What on earth are $P$ and $k$?
– user21820
yesterday
1
1
1) If you are going to delete a question, delete it. Don't just edit so it is unreadable. and 2) Don't delete questions. Just because you now know the answer doesn't mean other people will want to use it.
– fleablood
Nov 27 at 3:11
1) If you are going to delete a question, delete it. Don't just edit so it is unreadable. and 2) Don't delete questions. Just because you now know the answer doesn't mean other people will want to use it.
– fleablood
Nov 27 at 3:11
1
1
Do not vandalize your question to obfuscate it after posting. It's very rude to the person who bothered to answer.
– T. Bongers
Nov 27 at 4:31
Do not vandalize your question to obfuscate it after posting. It's very rude to the person who bothered to answer.
– T. Bongers
Nov 27 at 4:31
What on earth are $P$ and $k$?
– user21820
yesterday
What on earth are $P$ and $k$?
– user21820
yesterday
add a comment |
1 Answer
1
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votes
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0
down vote
accepted
$3^{k} cdot 3 < (k + 3)! cdot 3 < (k+3)!(k + 4) = (k + 4)! = ((k+1)+3)!$
Well $3 < 4 < 4 + k$
answered Nov 27 at 1:43
ZAF
4457
That prove P(k+1)
– ZAF
Nov 27 at 1:46
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
$3^{k} cdot 3 < (k + 3)! cdot 3 < (k+3)!(k + 4) = (k + 4)! = ((k+1)+3)!$
Well $3 < 4 < 4 + k$
answered Nov 27 at 1:43
ZAF
4457
That prove P(k+1)
– ZAF
Nov 27 at 1:46
add a comment |
up vote
0
down vote
accepted
$3^{k} cdot 3 < (k + 3)! cdot 3 < (k+3)!(k + 4) = (k + 4)! = ((k+1)+3)!$
Well $3 < 4 < 4 + k$
answered Nov 27 at 1:43
ZAF
4457
That prove P(k+1)
– ZAF
Nov 27 at 1:46
add a comment |
up vote
0
down vote
accepted
up vote
0
down vote
accepted
$3^{k} cdot 3 < (k + 3)! cdot 3 < (k+3)!(k + 4) = (k + 4)! = ((k+1)+3)!$
Well $3 < 4 < 4 + k$
answered Nov 27 at 1:43
ZAF
4457
$3^{k} cdot 3 < (k + 3)! cdot 3 < (k+3)!(k + 4) = (k + 4)! = ((k+1)+3)!$
Well $3 < 4 < 4 + k$
answered Nov 27 at 1:43
ZAF
4457
edited Nov 27 at 1:46
answered Nov 27 at 1:43
ZAF
4457
answered Nov 27 at 1:43
ZAF
4457
answered Nov 27 at 1:43
ZAF
4457
4457
That prove P(k+1)
– ZAF
Nov 27 at 1:46
add a comment |
That prove P(k+1)
– ZAF
Nov 27 at 1:46
That prove P(k+1)
– ZAF
Nov 27 at 1:46
That prove P(k+1)
– ZAF
Nov 27 at 1:46
add a comment |
1
1) If you are going to delete a question, delete it. Don't just edit so it is unreadable. and 2) Don't delete questions. Just because you now know the answer doesn't mean other people will want to use it.
– fleablood
Nov 27 at 3:11
1
Do not vandalize your question to obfuscate it after posting. It's very rude to the person who bothered to answer.
– T. Bongers
Nov 27 at 4:31
What on earth are $P$ and $k$?
– user21820
yesterday