Need help with Simplification step in Inductive Proof

  • Thread starter leo255
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  • #1
57
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Homework Statement



Let P (n) be the statement that 1^3 + 2^3 + · · · + n^3 = (n(n + 1)/2)^2 for the positive integer n. Prove inductively.

Homework Equations




The Attempt at a Solution


[/B]
I am skipping a few steps...I just need help here:

1/4K^2(k + 1)^2 + (k + 1)^3

Since I have access to the solution, the next step is this:

1/4(k+1)^2 [K^2 + 4 (k + 1)]

I am confused at how this is gotten to. I appreciate the help.

Thanks.
 

Answers and Replies

  • #2
35,222
7,040

Homework Statement



Let P (n) be the statement that 1^3 + 2^3 + · · · + n^3 = (n(n + 1)/2)^2 for the positive integer n. Prove inductively.

Homework Equations




The Attempt at a Solution


[/B]
I am skipping a few steps...I just need help here:

1/4K^2(k + 1)^2 + (k + 1)^3
Yes, you are skipping some steps.
The first term above is from ##1^3 + 2^3 + \dots + k^3## and the other term above is from adding ##(k + 1)^3## in your induction step.
In the two terms, do you notice that there is a common factor?
leo255 said:
Since I have access to the solution, the next step is this:

1/4(k+1)^2 [K^2 + 4 (k + 1)]

I am confused at how this is gotten to. I appreciate the help.

Thanks.
 
  • #3
57
2
Ahhh, yes. (k+1).

For some reason, I was trying to factor out k, k^2, etc.

So, from factoring out, I am getting:

1/4(k+1)^2 [k^2 + (k+1)]

I'm still a little confused as to where the 4, in front of the (k + 1) in the brackets is coming from. I assume it's because we are factoring out a 1/4, so we're just multiplying it by 4, to make it equal to one.
 
  • #4
35,222
7,040
Ahhh, yes. (k+1).

For some reason, I was trying to factor out k, k^2, etc.

So, from factoring out, I am getting:

1/4(k+1)^2 [k^2 + (k+1)]

I'm still a little confused as to where the 4, in front of the (k + 1) in the brackets is coming from. I assume it's because we are factoring out a 1/4, so we're just multiplying it by 4, to make it equal to one.
The part you left out is fouling you up.
The induction hypothesis is:
##1^3 + 2^3 + 3^3 + \dots + k^3 = \frac{k^2(k + 1)^2}{4}##
Now, work from the induction step:
##1^3 + 2^3 + 3^3 + \dots + k^3 + (k + 1)^3 = \frac{k^2(k + 1)^2}{4} + (k + 1)^3##

What happens when you factor out ##(k + 1)^2## from the right side?
 

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