# Toronto Math Forum

## MAT334--2020F => MAT334--Tests and Quizzes => Test 1 => Topic started by: Maria-Clara Eberlein on October 12, 2020, 05:16:22 PM

Title: 2018 test 1 variant C 2b
Post by: Maria-Clara Eberlein on October 12, 2020, 05:16:22 PM
How did we get from the second last to last line, in particular the calculation with (2n)! and (2n+2)! (See attachment)
Title: Re: 2018 test 1 variant C 2b
Post by: RunboZhang on October 13, 2020, 09:45:35 AM
It will be clearer if you expand the factorial:
$2n! = 2n \cdot (2n-1) \cdot ... \cdot 1$,
$(2n+2)! = (2n+2) \cdot (2n+1) \cdot (2n) \cdot ... \cdot 1 = (2n+2) \cdot (2n+1) \cdot (2n)!$
Thus $\frac {(2n)!}{(2n+2)!} = \frac {1}{(2n+2) \cdot (2n+1)}$
Title: Re: 2018 test 1 variant C 2b
Post by: bnogas on October 13, 2020, 11:34:38 AM
Don't we need to have 1/2n(2n+1) not 1/(2n+2)(2n+1) ?
Title: Re: 2018 test 1 variant C 2b
Post by: RunboZhang on October 13, 2020, 12:03:03 PM
I believe the equation can be written as:
$= |z| \cdot \frac{(n+1)^{3}}{(2n+2)(2n+1)}$
$= |z| \cdot \frac{(n+1)^{2}}{2 \cdot (2n+1)}$
Title: Re: 2018 test 1 variant C 2b
Post by: bnogas on October 13, 2020, 12:27:23 PM
Shouldn't it be 2n(n+1) in the denominator, not 2(n+1)?
Title: Re: 2018 test 1 variant C 2b
Post by: RunboZhang on October 13, 2020, 01:00:11 PM
No. I think it should be $(2n+2) \cdot (2n+1)$in the denominator. The last line of my equation in previous comment is the same stuff. Since (2n+2)=2(n+1), then cancel out the same term on denominator and numerator. If I didn't make myself clear pls comment below :)
Title: Re: 2018 test 1 variant C 2b
Post by: bnogas on October 13, 2020, 01:37:31 PM
Thx, but I meant in the very first post with a screenshot of the official solution it has 1/2n(2n+1),so I was just confused as to which one was the correct answer.
Title: Re: 2018 test 1 variant C 2b
Post by: RunboZhang on October 13, 2020, 01:51:14 PM
Sorry I didn't not make myself clear on this, I think the answer has a typo. The denominator in the original screenshot should be $(2n+2)\cdot(2n+1)$ :)
Title: Re: 2018 test 1 variant C 2b
Post by: Maria-Clara Eberlein on October 14, 2020, 03:53:24 PM
Ok if it has a typo then that makes sense, thank you!