# Homework Help: Properties of integers:math induction

1. Nov 17, 2008

### VanKwisH

1. The problem statement, all variables and given/known data
consider the following four equations:
1) 1=1
2) 2 + 3 + 4 = 1 + 8
3) 5 + 6 + 7 + 8 + 9 = 8 + 27
4) 10 + 11 + 12 + 13 + 14 + 15 + 16 = 27 + 64

conjecture the general formula suggested by these four equations and prove

2. Relevant equations

3. The attempt at a solution

I know the left side has n + (n+1) numbers,
and the right side has the last number occuring the in next equation,
i also see that the numbers on the right side come from an k^3 number.
but how would i write this in a general formula which would be sufficient to
satisfy all the equations? because to me it seems like a recursive definition but
i don't quite clearly understand how i would write out an answer for this....
and also what exactly am i supposed to write down to solve this??

2. Nov 17, 2008

### HallsofIvy

Notice also that the last number in the sum on the left is always a square, say, n2 and the last number on the right is n3. Use that as your base.
It looks to me like the first number on the left is (n-1)2+ 1 so the sum on the left is from (n-1)2+ 1 to n2.

3. Nov 17, 2008

### VanKwisH

so on the left side........ could i use sigma notation to verify the range and the addition of the numbers?? and on the right side ....... i can see that it's just like
1^3,
1^3 + 2^3 ,
2^3 + 3^3
3^3 + 4^3
how would i show this in terms of just using n ?
n + (n)^3 ?

the problem with this is that it will only apply to one specific instant of the equation,
or am i wrong??

4. Nov 17, 2008

### HallsofIvy

On both left and right side you are going to have to specify what "n" is!

5. Feb 25, 2010

### bmannaa

I guess the conjecture is
n^2 - 2 (n - 1) +....+ n^2 = (n-1)^3 + n^3
and it holds by trivial processing of the two sides

6. Feb 25, 2010

### Staff: Mentor

Does your guess work for the given equations?
There's a lot more to this than "trivial processing." Note the title of the thread.

7. Feb 25, 2010

### bmannaa

Yes it does
in the formula n^2 - 2 (n - 1) +......+ n^2 = (n-1)^3 + n^3
1) 1=1
put n = 1
2) 2 + 3 + 4 = 1 + 8
put n = 2
3) 5 + 6 + 7 + 8 + 9 = 8 + 27
put n = 3
4) 10 + 11 + 12 + 13 + 14 + 15 + 16 = 27 + 64
put n = 4
Yep! I didn't get to prove it by induction.