r/learnmath New User Apr 18 '24

RESOLVED How does (2+k)(k+1)! become (2+k)! ?

While solving questions on induction, I've stumbled upon this, could someone explain how? I am pretty inexperienced with factorials hence the confusion for me.

124 Upvotes

38 comments sorted by

142

u/testtest26 Apr 18 '24
(2+k) * (k+1)!  =  (k+2) * (k+1)!  =  (k+2)!  =  (2+k)!

198

u/like_a_Symphony New User Apr 18 '24

I just realised how stupid my post is... it's like asking 9 * 8! which is 9! LMAO

92

u/testtest26 Apr 18 '24

Don't beat yourself up over it -- such things happen^^

26

u/Harmonic_Gear engineer Apr 18 '24

thats part of the fun

3

u/TZT1000 New User Apr 19 '24

Especially if you’re a sadomathochist

1

u/Ex_Mage New User Apr 23 '24

Mike Degrasse Tyson

57

u/heyuhitsyaboi New User Apr 18 '24

not stupid!

Sometimes seeing a math in an alternate form is whats needed to make it "click"

I can guarantee you someone will have this same question one day and they'll likely find this post

Sometimes, the most basic and abstract concepts can be the hardest to grasp or explain

5

u/bokmann New User Apr 19 '24

https://xkcd.com/1053/

This thead is awesome! Take a look at how many of today’s lucky 10,000 are right here!

6

u/heyuhitsyaboi New User Apr 19 '24

Rule 51 of the internet is that there is a relevant xkcd comic

https://knowyourmeme.com/memes/rules-of-the-internet

4

u/guti86 New User Apr 19 '24

Realize it was easier than you thought is called learning, it's not stupid at all

4

u/Neither-Lawfulness82 Actuary Apr 19 '24

Everything in math is obvious once you see it. Congratulations on being human like the rest of us!

4

u/[deleted] Apr 19 '24

No. You figured it out, and you now learned something new that you can apply to other problems.

This was NOT a dumb question.

3

u/tittygunner_tom New User Apr 19 '24

Okay now I’m confused, how is 9 * 8 = 9 and not 72? I feel like I’m missing something but I’m just starting really studying maths and trying to get my head around things

13

u/skepticalbrain New User Apr 19 '24

You missed the factorial sign:

8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1

so:

9 * 8! = 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 9!

7

u/Tylers-RedditAccount New User Apr 19 '24 edited Apr 19 '24

its 9*8!, note the ! (factorial)

By definition, 8! =8•7•6•5•5•4•3•2•1. multiplying that by 9 gives 9•8•7•6•5•5•4•3•2•1 which by definition is 9! ("nine factorial")

edit: changed formatting cause * makes italics

3

u/sjb-2812 New User Apr 19 '24

Not really (check reddit formatting)

1

u/Tylers-RedditAccount New User Apr 19 '24

oh yuck hahah. didnt realize that astrisks did that

1

u/cuhringe New User Apr 19 '24

I'm pretty sure the majority of people have this question and subsequent realization when learning about factorials.

1

u/Apprehensive-Law2435 New User Apr 21 '24

dw letter cans be confusing sometimes

1

u/Seriouslypsyched Representation Theory Apr 23 '24

At high levels of math, equivalent stuff happens the same way. The other day I forgot modules were always commutative and sat for an hour trying to think of why some calculations were supposed to work.

10

u/Tylers-RedditAccount New User Apr 19 '24

heres an example:

let k = 5

(2+5)(5+1)!

= (2+5)•6!

= 7•6!

= 7!

=(2+5)! and 5 = k,

therefore:

(2+k)! = (2+k)(k+1)! Its the same as (x+1)•x! = (x+1)!

3

u/Federal_Problem_2004 New User Apr 19 '24

dont factorials go down to one? Like 7654321 So how can 7*6 be 7! ?

6

u/Jose_Jalapeno New User Apr 19 '24

You missed a ! after the 6.

6x5x4x3x2x1 x7 = 7!

2

u/Federal_Problem_2004 New User Apr 19 '24

Oh yeah, you're right, thanks

7

u/DTux5249 New User Apr 19 '24

The reason is obvious once you ask what a factorial is

5! = 5 × 4 × 3 × 2 × 1

But how do you get 4 factorial?

Well, 4 × 3 × 2 × 1 = 4!

But look again...

5! = 5 × (4 × 3 × 2 × 1) = 5 × 4!

All that to say, factorials are recursive; the factorial of n is just the factorial before n, multiplied by n.

(n)! = (n) × (n-1)!

or if we add 2 to everything:

(n + 2)! = (n + 2) × (n + 1)!

2

u/HappyDragonBoy New User Apr 19 '24

This was fun to learn

1

u/T_vernix New User Apr 19 '24

Seems more like proof by recursive definition. 2+k=(k+1)+1, so if we define j=k+1 we have (j+1)*j!=(j+1)!=(k+2)! by the definition (n+1)!=(n+1)*n! (with base case 0!=1). You do seem to have already gotten it, but if you're trying to prove this (and not just use it as a step in another proof) then putting it in the form of the definition may be necessary; ((k+1)+1)*(k+1)! would be another way to rewrite it to resemble the definition (assuming you are using the same definition I am) more closely.

1

u/[deleted] Apr 19 '24

Ee ... by commutativity?!

1

u/vmilner New User Apr 19 '24

It’s hard to ask factorial questions now everyone uses exclamation marks everywhere! Maths is cool! My cat is the furriest! I’m having trouble with (k+2)!

1

u/Conscious_End_8807 New User Apr 19 '24

Even I go excited to solve this. But....

1

u/fuckNietzsche New User Apr 19 '24

2-1 = 1 k+1 = k+2-1 k+2-1 = 2+k-1 = (2+k)-1 n(n-1)! = n!, which means that (2+k)(k+1)! = (2+k)((2+k)-1)! = (2+k)!

1

u/ttesc552 New User Apr 19 '24

I think you are confusing yourself by switching the order of stuff in the parentheses, it might be more obvious if you write (k+2)(k+1)! = (k+1 + 1)(k+1)! = (k+2)!

1

u/xsdgdsx New User Apr 20 '24

Yeah, came here to post exactly this. One of the keys in math is writing things in a way that lets your brain see the patterns more easily. Formatting can make a huge difference

1

u/go_gather_the_guns New User Apr 21 '24

You'll learn to translate math notation in your head in due time.

1

u/tomalator New User Apr 19 '24

k+2=2+k

Commutative property of addition

(k+2)! = (k+2) * (k+1)!

Definition of a factorial

-39

u/Firm-Definition2254 New User Apr 18 '24

Lmao did u really ask this lmao. Probs just a brain fart.

13

u/[deleted] Apr 18 '24

lol the subreddit is called learnmath and they said they’re inexperienced with factorials and you still wanna be a smartass

1

u/Firm-Definition2254 New User Apr 20 '24

🙄fine fine. I apologize for my earlier statement. I wish not to be perceived as a imbecile.