Mathematical Induction (1 Viewer)
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(i) ln(6!)>6 so true for S(6)
Assume S(n) true. ie. ln(n!)>n
Consider S(n+1):
ln(n+1)!
=ln[(n+1)n!]
=ln(n+1)+ln(n!)
>ln(n+1)+n
>n+1 since ln(n+1)>1 for n>6
S(n+1) is true if S(n) is true but S(6) is true so it is true for n>=6
(ii) ln(n!)>n
n!>e^n
(1/(n!))<(1/e^n)
(iii) [(1/1!)+(1/2!)+(1/3!)+(1/4!)+(1/5!)] + (1/6!)+...
<[103/60] + (1/e^6) + (1/e^7) + ...
<[103/60] + (1/e^6)/(1-(1/e)) using limiting sum of GP starting term (1/e^6)
=(103/60) + 1/(e^5(e-1))
Assume S(n) true. ie. ln(n!)>n
Consider S(n+1):
ln(n+1)!
=ln[(n+1)n!]
=ln(n+1)+ln(n!)
>ln(n+1)+n
>n+1 since ln(n+1)>1 for n>6
S(n+1) is true if S(n) is true but S(6) is true so it is true for n>=6
(ii) ln(n!)>n
n!>e^n
(1/(n!))<(1/e^n)
(iii) [(1/1!)+(1/2!)+(1/3!)+(1/4!)+(1/5!)] + (1/6!)+...
<[103/60] + (1/e^6) + (1/e^7) + ...
<[103/60] + (1/e^6)/(1-(1/e)) using limiting sum of GP starting term (1/e^6)
=(103/60) + 1/(e^5(e-1))
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lyounamu
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1.Hi,
Could someone please do this for me? Also, does anyone know where this question is from?
Thanks
When n=6,
LHS = ln(6!) = 6.579....
RHS = 6
Therefore, LHS > RHS
Proved true when n=6
Now assume that n=k
so ln(k!) > k
Prove that it is true for n=k+1 (and k = t-5) where k>=6
i.e. LHS = ln((k+1)!) = ln ((k+1)(k)!) = ln(k+1) + ln(k!) > k + ln (k+1) > k + 1 = LHS (since ln(k+1) > 1 for every value for k)
proved true by the principle of mathematical induction
2.
(ii) ln(n!) > n (proved above)
n! > e^n
1/(n!) < 1/e^n
3.
From above, 1/1! + 1/2! + 1/3! + 1/4! +1/5! = 103/60 ..(1)
and the rest of 1/6! + 1/7! + ... + < 1/e^6 + 1/e^7 + 1/e^8... (geometric series to infinity) = 1/e^6 x (1/e-1) = 1/(e^5(e-1)) ..(2)
so 1/1! + 1/2! + 1/3! + 1/4! +1/5! + ... < 103/60 + 1/(e^5(e-1)) (because in the first they were equal but in the second one was higher)
Last edited:
GUSSSSSSSSSSSSS
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yeh good solution
lol when i did it i wasnt gettin the 103/60 on the RHS in (iii)
and it was pissing me off
then i realised it was for n>6 and i was happy XDD lol
lyounamu
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rule number 1: never happy over a maths question :rofl:
GUSSSSSSSSSSSSS
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hahahaha sorry i shall correct myself
"i felt a little better, but still heavily pissed off at the fact that i was doing math"
is that better? lol
lyounamu
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So why are you such a tool?
lyounamu
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lol you are the one who posted up the question..
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I've confirmed that it is from a past CSSA paper...I've seen that question before. It's also in the Cambridge Study guide if anyone has it. To be honest, the last part of that question is just stupid...lol
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It came from the very last question in the CSSA 2003:
http://www.boredofstudies.org/courses/maths/3u/2003_Mathematics_Extension_1_T_Trial_CSSA.pdf
http://www.boredofstudies.org/courses/maths/3u/2003_Mathematics_Extension_1_T_Trial_CSSA.pdf
GUSSSSSSSSSSSSS
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lol and thats the part where i took longest on
i was like wtf where this 103/60 come from wtf going on
then realised lol
*embarrasing*
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<TABLE class=tborder style="BORDER-TOP-WIDTH: 0px" cellSpacing=1 cellPadding=3 width="100%" align=center border=0><TBODY><TR title="Post 4268508" vAlign=top><TD class=alt2 align=middle width=125>2 posts later...</TD><TD class=alt1>
<HR style="COLOR: #d1d1e1; BACKGROUND-COLOR: #d1d1e1" SIZE=1>Yeah, it's from last year's half yearly examination paper (my school's) but I have seen it in other trials, so I'm sure the teacher got it from somewhere.
Apparantly like nobody got it correct, lol. </TD></TR></TBODY></TABLE>
I wonder, Why did you ask if it was a CSSA question if you already knew it was?
clintmyster
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he didn't say it was a CSSA question, just that he saw it other papers and presumed that it must've come from some company that several schools would have (e.g. Neap or CSSA in this case)
MC Squidge
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ohhhh onlooker got shut the fuck down
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Unfortunately not ^ Faggot 
<TABLE class=tborder style="BORDER-TOP-WIDTH: 0px" cellSpacing=1 cellPadding=3 width="100%" align=center border=0><TBODY><TR title="Post 4273929" vAlign=top><TD class=alt2 align=middle width=125>jm01</TD><TD class=alt1>Re: Mathematical Induction
<HR style="COLOR: #d1d1e1; BACKGROUND-COLOR: #d1d1e1" SIZE=1>Onlooker, do you even know what 'CSSA' means/stands for, and what they do?
Because if you do, you would've found my question perfectly valid and reasonable. In future stay out of my posts. </TD></TR></TBODY></TABLE>
CBF quoting your sorry ass. 1. stay out of your posts? HAHA.
2. I know what CSSA means thankyou
3. Do not get a smart mouth with me, because you won't hear the end of it
<TABLE class=tborder style="BORDER-TOP-WIDTH: 0px" cellSpacing=1 cellPadding=3 width="100%" align=center border=0><TBODY><TR title="Post 4273929" vAlign=top><TD class=alt2 align=middle width=125>jm01</TD><TD class=alt1>Re: Mathematical Induction
<HR style="COLOR: #d1d1e1; BACKGROUND-COLOR: #d1d1e1" SIZE=1>Onlooker, do you even know what 'CSSA' means/stands for, and what they do?
Because if you do, you would've found my question perfectly valid and reasonable. In future stay out of my posts. </TD></TR></TBODY></TABLE>
CBF quoting your sorry ass. 1. stay out of your posts? HAHA.
2. I know what CSSA means thankyou
3. Do not get a smart mouth with me, because you won't hear the end of it