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Year 12s: what will be the first way you research your future options? >>

Year 12s: what will be the first way you research your future options? >>

common denominator of a factorial

whats the lowest common denominator of this factorial and why? thanks

this isn't equivalent to: 1/x - 1/(x+1)

so how would I go about it?

(edited 6 years ago)

Screen Shot 2017-11-10 at 14.51.26.png17.9KB

Write the factorials out explicitly and it should become obvious.

OP

Original post by IrrationalRoot

Write the factorials out explicitly and it should become obvious.

I did try that at first but now I see it :smile:

:smile:

OP

Original post by IrrationalRoot

Write the factorials out explicitly and it should become obvious.

so, I see how the lowest common denominator is (r+1)!

however, I dont understand the first fraction. how does: r!(r+1) = (r+1)!?

(edited 6 years ago)

Screen Shot 2017-11-10 at 15.08.03.png31.3KB

Study Forum Helper

Original post by Maths&physics

so, I see how the lowest common denominator is (r+1)!

however, I dont understand the first fraction. how does: r!(r+1) = (r+1)!?

\displaystyle n! = n \cdot \underbrace{(n-1) \cdot (n-2) \cdot ... \cdot 2 \cdot 1}_{= (n-1)!}

OP

Original post by RDKGames

\displaystyle n! = n \cdot \underbrace{(n-1) \cdot (n-2) \cdot ... \cdot 2 \cdot 1}_{= (n-1)!}

sorry but I dont get what you wrote.

r!(r+1) = (r+1)!

1(1+1) x 2(2+1) x 3(3+1) = (1+1) x (2+1) x (3+1)

2 x 6 x 12 = 2 x 3 x 4 ????

(edited 6 years ago)

Study Forum Helper

(Ok the app is being useless so I cant quote you properly)

Anyway, what do you think ‘!’ means?

(edited 6 years ago)

Original post by Maths&physics

sorry but I dont get what you wrote.

r!(r+1) = (r+1)!

1(1+1) + 2(2+1) + 3(3+1) = (1+1) + (2+1) + (3+1)

2 + 6 + 12 = 2 + 3 + 4 ????

You're overcomplicating this, it's just:

r!(r+1)=(1\times 2\times 3\times \ldots \times r) \times (r+1)= 1\times 2\times 3\times \ldots \times r \times (r+1) = (r+1)!

.

As you can see it follows immediately from the definition of factorial (actually it IS the definition of factorial, technically speaking).

OP

Original post by RDKGames

(Ok the app is being useless so I cant quote you properly)

Anyway, what do you think ‘!’ means?

I thought ! = 1x2x3x4, etc?

Study Forum Helper

Original post by Maths&physics

I thought ! = 1x2x3x4, etc?

Yes.... n!=n(n-1)(n-2)...(2)(1) as I said in my initial post.

You didn’t apply this definition in your flawed example

(edited 6 years ago)

OP

Original post by IrrationalRoot

You're overcomplicating this, it's just:

r!(r+1)=(1\times 2\times 3\times \ldots \times r) \times (r+1)= 1\times 2\times 3\times \ldots \times r \times (r+1) = (r+1)!

.

As you can see it follows immediately from the definition of factorial (actually it IS the definition of factorial, technically speaking).

so its:

(1x2x3x4....)(r+1) = (1+1)(1+2)(1+3)(1+4)....

(1x2x3x4....)(1+1)(2+1)(3+1)(4+1) = (1+1)(1+2)(1+3)(1+4)....

I dont see how theyre equivalent?

OP

Original post by IrrationalRoot

You're overcomplicating this, it's just:

r!(r+1)=(1\times 2\times 3\times \ldots \times r) \times (r+1)= 1\times 2\times 3\times \ldots \times r \times (r+1) = (r+1)!

.

As you can see it follows immediately from the definition of factorial (actually it IS the definition of factorial, technically speaking).

ah, do you automatically multiply whats inside the bracket by 1x2x3.... when there is an ! besides it? :smile:

:smile:

so r! = (1x2x3....)(1x2x3...)?

(edited 6 years ago)

OP

Original post by RDKGames

Yes.... n!=n(n-1)(n-2)...(2)(1) as I said in my initial post.

You didn’t apply this definition in your flawed example

I didn't: sorry

(edited 6 years ago)

Study Forum Helper

Original post by Maths&physics

so its:

(1x2x3x4....)(r+1) = (1+1)(1+2)(1+3)(1+4)....

(1x2x3x4....)(1+1)(2+1)(3+1)(4+1) = (1+1)(1+2)(1+3)(1+4)....

I dont see how theyre equivalent?

Original post by Maths&physics

ah, do you automatically multiply whats inside the bracket by 1x2x3.... when there is an ! besides it? :smile:

:smile:

so r! = !

its the same thing?

??????????

No! We are clearly saying that whenever you see something with a ! right after it, that means you multiply every integer from 1 up to it together..

Original post by Maths&physics

ah, do you automatically multiply whats inside the bracket by 1x2x3.... when there is an ! besides it? :smile:

:smile:

so r! = !

its the same thing?

1!=1

2!=1\times 2

3!=1 \times 2 \times 3

4!=1 \times 2 \times 3 \times 4

r!=1 \times 2 \times 3 \times 4 \times \ldots \times (r-1) \times r

You really can't go wrong with this.

OP

Original post by RDKGames

??????????

No! We are clearly saying that whenever you see something with a ! right after it, that means you multiply every integer from 1 up to it together..

so r! = (1x2x3..)(1x2x3...)?

Original post by Maths&physics

I didn't: sorry

You seem confused in these posts. Taking it back to basics do you know what

5!

means? And do you understand why these are true?

5! = 5 \times 4!

5! = 5\times 4 \times 3!

I think it's best that you understand this before introducing variables.

Study Forum Helper

Original post by Maths&amp

so r! = (1x2x3..)(1x2x3...)?

Just open up your textbook and look at how it’s defined and used.

OP

OK, I'VE FINALLY GOT IT! thanks :smile:

:smile:

:biggrin:

OP

Original post by IrrationalRoot

You're overcomplicating this, it's just:

r!(r+1)=(1\times 2\times 3\times \ldots \times r) \times (r+1)= 1\times 2\times 3\times \ldots \times r \times (r+1) = (r+1)!

.

As you can see it follows immediately from the definition of factorial (actually it IS the definition of factorial, technically speaking).

ive got it.

r = 3

3!(3+1) = (3+1)!

I just realised you told me that about an hour ago! :biggrin:

:biggrin:

thanks

(edited 6 years ago)

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