Question

Prove that P(n) : n(n+1)(n+5) is divisible by 3

Answer 1

@waterineyes

Answer 2

What are our tool? Are you trying to do this with induction? or do we have modular arithmetic at our disposal?

Answer 3

induction

Answer 4

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.

Answer 5

@beginnersmind i didn't get it :/ sorry

can u explain it step by step ?

Answer 6

According to induction:

Put n = 1 and see are you getting what the question says..

Answer 7

@beginnersmind do not stop.

Carry on..

I am just trying as @ashna is doing..

Answer 8

i know that water tell me from ,

to prove p(k+1) is true

Answer 9

My answer didn't use induction so I'd rather not go into a long explanation.

Answer 10

Ha ha ha ha...


You knew that??


Just kidding..

Answer 11

we get k = 3M/ (K+1)(K+5) Right ?

Answer 12

Replace n by k+1 first...

Answer 13

yeah did , then ?

Answer 14

Then Look carefully it will also be divisible by 3..

Ha ha ha ha...

Answer 15

c'mon Water i don't understand :I

Answer 16

\[= (k+1)(k+2) (k+6)\]

Answer 17

okay

Answer 18

Really??

Answer 19

where r yu goin to substitute k = 3M/ (K+1)(K+5) ?

Answer 20

Wait...

Answer 21

okay

Answer 22

It is now 6 when I studied Induction..

Answer 23

*6 years..

Answer 24

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.

Answer 25

yeah .. on assuming i got k = 3M/ (K+1)(K+5)

3rd step am stuck :I

Answer 26

what does the M stand for?

Answer 27

Ah, ok, see what you did there. You said there's a number M such that P(k) = 3M

Answer 28

M = divisible by 3

Answer 29

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?

Answer 30

yes :)

Answer 31

Cool :)

To check, what did you get for P(k+1) - P(k) ?

Answer 32

What if we find the value of k+1 from the assumption??

Answer 33

\[k+1 = \frac{3M}{k (k+5)}\]