Mathematical Induction (1 Viewer)

cutemouse · Apr 15, 2009

Hi,

Could someone please do this for me? Also, does anyone know where this question is from?

Thanks

azureus88 · Apr 15, 2009

(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))

lyounamu · Apr 15, 2009

jm01 said:
Hi,

Could someone please do this for me? Also, does anyone know where this question is from?

Thanks

1.

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)

GUSSSSSSSSSSSSS · Apr 15, 2009

azureus88 said:
(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))

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 · Apr 15, 2009

GUSSSSSSSSSSSSS said:
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

rule number 1: never happy over a maths question :rofl:

GUSSSSSSSSSSSSS · Apr 15, 2009

lyounamu said:
rule number 1: never happy over a maths question :rofl:

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 · Apr 15, 2009

GUSSSSSSSSSSSSS said:
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

-Onlooker- · Apr 15, 2009

GUSSSSSSSSSSSSS said:
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

So why are you such a tool?

cutemouse · Apr 15, 2009

Thanks,

Is this a CSSA question?

lyounamu · Apr 15, 2009

jm01 said:
Thanks,

Is this a CSSA question?

lol you are the one who posted up the question..

cutemouse · Apr 15, 2009

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.

Trebla · Apr 15, 2009

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

cutemouse · Apr 15, 2009

Hi Trebla, Thanks for your info. Do you know which year it's from?

Also, what do you think of the Cambridge Study in terms of its usefulness?

Trebla · Apr 15, 2009

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

GUSSSSSSSSSSSSS · Apr 15, 2009

Trebla said:
the last part of that question is just stupid...lol

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*

-Onlooker- · Apr 17, 2009

jm01 said:
Thanks,

Is this a CSSA question?

<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 · Apr 17, 2009

-Onlooker- said:
<table class="tborder" style="border-top-width: 0px;" width="100%" align="center" border="0" cellpadding="3" cellspacing="1"><tbody><tr title="Post 4268508" valign="top"><td class="alt2" width="125" align="middle">2 posts later...</td><td class="alt1">
<hr style="color: rgb(209, 209, 225); background-color: rgb(209, 209, 225);" 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?

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)

cutemouse · Apr 17, 2009

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.

MC Squidge · Apr 19, 2009

ohhhh onlooker got shut the fuck down

-Onlooker- · May 28, 2009

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

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