ashna
Prove that P(n) : n(n+1)(n+5) is divisible by 3
Delete
Share
This Question is Closed
ashna
Best Response
You've already chosen the best response.
0
@waterineyes
joemath314159
Best Response
You've already chosen the best response.
0
What are our tool? Are you trying to do this with induction? or do we have modular arithmetic at our disposal?
ashna
Best Response
You've already chosen the best response.
0
induction
beginnersmind
Best Response
You've already chosen the best response.
3
My favourite would be to write it as n(n+1)(n+2) + 3n(n+1)
The second term is divisible by 3 and the first one is the product of 3 consecutive integers therefore also divisible by 3. The sum of 2 numbers both divisible by 3 is also divisible by 3.
ashna
Best Response
You've already chosen the best response.
0
@beginnersmind i didn't get it :/ sorry
can u explain it step by step ?
waterineyes
Best Response
You've already chosen the best response.
0
According to induction:
Put n = 1 and see are you getting what the question says..
waterineyes
Best Response
You've already chosen the best response.
0
@beginnersmind do not stop.
Carry on..
I am just trying as @ashna is doing..
ashna
Best Response
You've already chosen the best response.
0
i know that water tell me from ,
to prove p(k+1) is true
beginnersmind
Best Response
You've already chosen the best response.
3
My answer didn't use induction so I'd rather not go into a long explanation.
waterineyes
Best Response
You've already chosen the best response.
0
Ha ha ha ha...
You knew that??
Just kidding..
ashna
Best Response
You've already chosen the best response.
0
we get k = 3M/ (K+1)(K+5) Right ?
waterineyes
Best Response
You've already chosen the best response.
0
Replace n by k+1 first...
ashna
Best Response
You've already chosen the best response.
0
yeah did , then ?
waterineyes
Best Response
You've already chosen the best response.
0
Then Look carefully it will also be divisible by 3..
Ha ha ha ha...
ashna
Best Response
You've already chosen the best response.
0
c'mon Water i don't understand :I
waterineyes
Best Response
You've already chosen the best response.
0
\[= (k+1)(k+2) (k+6)\]
ashna
Best Response
You've already chosen the best response.
0
okay
waterineyes
Best Response
You've already chosen the best response.
0
Really??
ashna
Best Response
You've already chosen the best response.
0
where r yu goin to substitute k = 3M/ (K+1)(K+5) ?
waterineyes
Best Response
You've already chosen the best response.
0
Wait...
ashna
Best Response
You've already chosen the best response.
0
okay
waterineyes
Best Response
You've already chosen the best response.
0
It is now 6 when I studied Induction..
waterineyes
Best Response
You've already chosen the best response.
0
*6 years..
beginnersmind
Best Response
You've already chosen the best response.
3
Ok, this is how you do it with induction.
First prove it for n =1 (plug it in and check if it's divisible by 3)
Second assume that it's true for P(k). Using this try to prove it's also true for P(k+1)
In this case I would try to prove that P(k+1) - P(k) is divisible by 3.
ashna
Best Response
You've already chosen the best response.
0
yeah .. on assuming i got k = 3M/ (K+1)(K+5)
3rd step am stuck :I
beginnersmind
Best Response
You've already chosen the best response.
3
what does the M stand for?
beginnersmind
Best Response
You've already chosen the best response.
3
Ah, ok, see what you did there. You said there's a number M such that P(k) = 3M
ashna
Best Response
You've already chosen the best response.
0
M = divisible by 3
beginnersmind
Best Response
You've already chosen the best response.
3
Ok, I'd do it slightly differently. I'd prove that the difference of P(k+1) and P(k) is divisible by 3.
Then using this and the induction hypothesis it follows that P(k+1) is also divisible by 3.
Does that make sense?
ashna
Best Response
You've already chosen the best response.
0
yes :)
beginnersmind
Best Response
You've already chosen the best response.
3
Cool :)
To check, what did you get for P(k+1) - P(k) ?
waterineyes
Best Response
You've already chosen the best response.
0
What if we find the value of k+1 from the assumption??
waterineyes
Best Response
You've already chosen the best response.
0
\[k+1 = \frac{3M}{k (k+5)}\]
ashna
Best Response
You've already chosen the best response.
0
|dw:1355641516686:dw|
waterineyes
Best Response
You've already chosen the best response.
0
Hey I don't know how to prove that, I am just giving my Ideas which can be useless too..
waterineyes
Best Response
You've already chosen the best response.
0
Now this does not look like if it is divisible by 3 or not??
ashna
Best Response
You've already chosen the best response.
0
yeah .. :P
waterineyes
Best Response
You've already chosen the best response.
0
Wait, do not trust me..
Please confirm it from good source, may be I am wrong because I am not sure..
ashna
Best Response
You've already chosen the best response.
0
@beginnersmind suggest :)
beginnersmind
Best Response
You've already chosen the best response.
3
I don't think we are going in the right direction :(
Let's just start by checking if the statement holds for n = 1
ashna
Best Response
You've already chosen the best response.
0
okay !
beginnersmind
Best Response
You've already chosen the best response.
3
And by let us, I mean you ;)
waterineyes
Best Response
You've already chosen the best response.
0
@UnkleRhaukus help here please..
beginnersmind
Best Response
You've already chosen the best response.
3
Let me rephrase:
What's the value of P(1)?
ashna
Best Response
You've already chosen the best response.
0
12
ashna
Best Response
You've already chosen the best response.
0
divisible by 3 , so true !
beginnersmind
Best Response
You've already chosen the best response.
3
Ok. So we've proven that it holds for n =1.
Now, what's
P(k)?
P(k+1)?
ashna
Best Response
You've already chosen the best response.
0
p(k) = 3M / (k+1)(k+5)
beginnersmind
Best Response
You've already chosen the best response.
3
Let's forget about the induction hypothesis, just express P(k) and P(k+1) using the original definition
waterineyes
Best Response
You've already chosen the best response.
0
She has assumed that p(k) is divisible by 3 so 3M where M is an Integer..
ashna
Best Response
You've already chosen the best response.
0
i dont know how to do other than that :I
@beginnersmind
beginnersmind
Best Response
You've already chosen the best response.
3
I mean, the same way you plugged in n = 1 to get P(1) you can plug in k to get P(k)
waterineyes
Best Response
You've already chosen the best response.
0
Just replace n by k there..
ashna
Best Response
You've already chosen the best response.
0
okay so k(k+1)(k=5) = divisible by 3
yu meant this ?
waterineyes
Best Response
You've already chosen the best response.
0
p(k) = .......................
beginnersmind
Best Response
You've already chosen the best response.
3
I mean P(k) = k(k+1)(k+5)
We'll use the fact that it's divisible by 3 later, but not yet.
ashna
Best Response
You've already chosen the best response.
0
okay , now how to proceed ?
beginnersmind
Best Response
You've already chosen the best response.
3
So the same way, what is P(k+1)?
ashna
Best Response
You've already chosen the best response.
0
P(k+1) = (k+1)(k+2)(k+6)
beginnersmind
Best Response
You've already chosen the best response.
3
right
ashna
Best Response
You've already chosen the best response.
0
and now ?
beginnersmind
Best Response
You've already chosen the best response.
3
We want to look at P(k+1) - P(k)
ashna
Best Response
You've already chosen the best response.
0
why ?
beginnersmind
Best Response
You've already chosen the best response.
3
This is the difference between two consecutive terms. For example the difference between the second term and the first term is P(2)-P(1)
ashna
Best Response
You've already chosen the best response.
0
okay
beginnersmind
Best Response
You've already chosen the best response.
3
If P(2) - P(1) is divisible by 3 and P(1) is divisible by 3 then so is P(2). Do you see why?
ashna
Best Response
You've already chosen the best response.
0
yes i do
beginnersmind
Best Response
You've already chosen the best response.
3
Let's try to write out P(k+1) - P(k) and see if it's divisible by 3
beginnersmind
Best Response
You've already chosen the best response.
3
You already calculated P(k+1) and P(k) separately so just take the difference and see if you can write it in some nice way.
waterineyes
Best Response
You've already chosen the best response.
0
It p(1) = 6
p(2) = 8
Then : p(2) - p(1) = 2
ashna
Best Response
You've already chosen the best response.
0
okay i've got (k+1)[(3k+12)]
waterineyes
Best Response
You've already chosen the best response.
0
This is divisible by 3 now...
beginnersmind
Best Response
You've already chosen the best response.
3
That's nice. I'd rewrite it as 3(k+1)(k+4) just to emphasize that it's divisible by 3.
ashna
Best Response
You've already chosen the best response.
0
right YAY :D
ty @beginnersmind and @waterineyes
beginnersmind
Best Response
You've already chosen the best response.
3
Not finished yet.
But almost :)
ashna
Best Response
You've already chosen the best response.
0
i can do the rest ty :)
beginnersmind
Best Response
You've already chosen the best response.
3
Ok :)