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Math Induction

I am trying to figure out a problem and need some help.
The problem is listed below.

Using induction, verify that each equation is true for every positive integer n.
1 · 2 + 2 · 3 + 3 · 4 + ......+ n ( n + 1 ) =       (n (n + 1)(n + 2) ) / 3

I can get the number 1 to work for n but numbers above that I cannot. Any help would be appreciated.
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rcanter
Asked:
rcanter
2 Solutions
 
ozoCommented:
Assuming it works for n (e.g. 1) you need to prove it works for n+1
Can you find the difference between the sum for n and the sum for n+1?
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rcanterAuthor Commented:
I cannot get n+1 to work, although I may be doing something wrong. I do not understand induction much and this is the first time I have tried, although it has been a long night and day of trying :-(
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JR2003Commented:
You need to show that
((n+1)(n + 2)(n + 3) ) / 3     is equal to     (n (n + 1)(n + 2) ) / 3 + ((n+1)(n+2))

=(n (n + 1)(n + 2) ) / 3 + (3(n+1)(n+2))/3

which simplifies to:
((n+1)(n + 2)(n + 3) ) / 3
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Harisha M GCommented:
The principle of mathematical induction is very simple. You show that it works for atleast one number (generally 1) and then prove that if it works for n, it works for n+1 too.

That way, if it works for 1, then it should work for 1+1 = 2 also.
Since it works for 2, it should work for 2+1 = 3 also, and so on..

You said you can find that it works for 1. So, I will show the other part...

1 · 2 + 2 · 3 + 3 · 4 + ......+ n ( n + 1 ) =      (n (n + 1)(n + 2) ) / 3

Now, you need to add the "next term of the last term" of the LHS to both the sides..
Last term of LHS = n(n+1)
Next term (Change n to n+1, that's all)  = (n+1)(n+1 + 1) = (n+1)(n+2)

So, adding it to both the sides,

1 · 2 + 2 · 3 + 3 · 4 + ......+ n(n+1) + (n+1)(n+2) =  (n(n+1)(n+2))/3 + (n+1)(n+2)

Which is nothing but..

1 · 2 + 2 · 3 + 3 · 4 + ...... + (n+1)(n+2) =  (n(n+1)(n+2))/3 + 3(n+1)(n+2)/3
                        = (n+1)(n+2)(n+3)/3

You can see that the RHS is nothing but the original RHS, but n being replaced by n+1.

That proves that if it works for n, it must work for n+1 too..
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