Mathematical Induction (1 Viewer)

[cutemouse]

Hi,

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

Thanks

 

[azureus88]

(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]

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]

(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]

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]

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]

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-]

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]

Thanks,

Is this a CSSA question?

 

[lyounamu]

Thanks,

Is this a CSSA question?

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

 

[cutemouse]

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]

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]

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]

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]

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-]

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]

<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]

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]

ohhhh onlooker got shut the fuck down

 

[-Onlooker-]

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