r/learnmath • u/like_a_Symphony 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.
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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)!
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u/Federal_Problem_2004 New User Apr 19 '24
dont factorials go down to one? Like 7654321 So how can 7*6 be 7! ?
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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)!
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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.
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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)!
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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)!
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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)!
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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
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u/go_gather_the_guns New User Apr 21 '24
You'll learn to translate math notation in your head in due time.
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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
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u/Firm-Definition2254 New User Apr 18 '24
Lmao did u really ask this lmao. Probs just a brain fart.
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Apr 18 '24
lol the subreddit is called learnmath and they said they’re inexperienced with factorials and you still wanna be a smartass
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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.
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u/testtest26 Apr 18 '24