help on induction! (2 Viewers)

[r.haidar]

how woulld u d this question? its q7 b) on 2003 CSSA trial

[img=http://img217.imageshack.us/img217/1807/image10xl.png]

 

[AreYouAlright?]

Personally, i'd leave it and move on and hope no one else in the state can do it. However, my tactics don't always work out

 

[rama_v]

http://community.boredofstudies.org/showthread.php?t=79236

 

[Pace_T]

can someone please explain the first two parts i don't get them
thanks.

 

[rama_v]

can someone please explain the first two parts i don't get them
thanks.

(1) prove for n=6
ln 6! > 6
6.57>6
true

(2) assueme n=k
ln n! - k >0

(3) prove true for n=k+1
i.e. prove ln(k+1)!>k+1

LHS = ln(k+1)!
= lnk! +ln(k+1)

but ln k! + ln(k+1) > ln(k+1) +k
for k>6, k+1 >e
ln (k+1) > ln e since ln e = 1
.: ln[(k+1)!]>1+k

Hence if the result is true for n=k it has been proven true for n=k+1, so it is true for all n where n>6, proven by mathematical induction...

Full solutions: http://www.boredofstudies.org/cours...matics_Extension_1_T_Trial_Solutions_CSSA.pdf


so therefore ln n! >k for all positve integers n>6 proven true by mathematical induction

For the second part:
ln n! > n
e^{ln n!} > eⁿ
1/e^{ln n!} < 1/eⁿ
so 1/n! < 1/eⁿ

 

[Pace_T]

thanks ....