Question

Compute: \[\log_2 (\frac{2012!!}{1006!})\] NO CALCULATORS PERMITTED!

Answer 1

\[{2012 \over 1006} = 2 \]

Answer 2

\[ \log_2 2 = ?\]

Answer 3

Unless that is really a factorial.

Answer 4

Yup...By definition:
\[n! = n!(n - 1)!(n - 2)!...1!\]\[n!! = n!(n - 2)!(n - 4)!...1! or 2!\]based on whether it's even or odd.

Answer 5

That's sad.

Answer 6

Oops. No ! marks in between lol

Answer 7

Take 2 common in the numerator..

\(2^{1006}\) (1006)!

May be this will help..

Answer 8

I'm not quite understanding what you mean?

Answer 9

I show you..

Answer 10

\[2012!! = (2012)(2010)(2008)(2006)(2004)....................2!\]

As these are in multiplication so you have to take 2 common from all the terms:

\[2012!! = 2^{1006}(1006)(1005)(1004)(1003)(1002)....................1!\]

\[2012!! = 2^{1006} \times 1006!\]

Answer 11

I am having doubt that should be 1006 or 1005 in the power..

Answer 12

Oh...so you're dividing each of the parts of the factorial by 2?

Answer 13

I am taking common (distributive property)..

Answer 14

So then:
\[\log_2(\frac{2012!!}{1006!}) \implies \log_2(\frac{2^{1006} \times 1006!}{1006!}) \implies \log_2(2^{1006}) \implies 1006\]Is that what happens?

Answer 15

(2012)(2010)(2008)

Take 2 common from first:

2(1006) (2010) (2008)

Like wise take 2 common from all terms:

2(1006) 2(1005) 2(1004) -------------2^3 (1006)(1005)(1004)

Answer 16

Like wise..

Answer 17

It's because all of the terms are even, so you can remove a 2 from each which when multiplied to gether form \(2^{1006}\)since there are 1006 terms in 2012!?

Answer 18

*factor out

Answer 19

Yes that is what I am saying..

But I am in a doubt that will be 1005 or 1006..

Answer 20

Let me think for two minutes..

Answer 21

I think it would be 1006 since the biggest term 2012, when divided by 2 forms 1006 which the rest follow suit...plus, if you know the context of this problem, there wouldn't be a way to get the end result to be pretty...

Answer 22

Yes it will be 1006 using AP..

Answer 23

Wait, what does AP stand for?

Answer 24

2 + 4 + 6 + 2012

Find n using Arithmetic Progression (A.P).

Answer 25

They are total 1006 terms so use will have factor out or divide 2 1006 times..

Answer 26

Ah ok...thank you! I understand it now! :)

Answer 27

Welcome dear..