Help with simple proof by mathematical induction

In this case, you need to factor out 1/6 and then simplify what is left in the brackets. This will give you the desired result. In summary, to prove the given equation by induction, you need to manipulate the induction step by factoring out 1/6 and simplifying the remaining terms in the brackets.
  • #1
mcraze123
1
0

Homework Statement



prove:
0^2 + 1^2 + 2^2 + ... + n^2 = n(n+1)(2n+1)/6


Homework Equations





The Attempt at a Solution



I'm confused on how to prove this by induction. I'm not exactly sure what the goal of the rearrangement is after substituting (n+1). Any help is much appreciated!

base case: n = 0
0^2 = 0(0+1)(2*0+1)/6

induction step:
(0^2+1^2+2^2+...+n^2) + (n+1)^2 = (n+1)((n+1)+1)(2(n+1)+1)/6

n(n+1)(2n+1)/6 + (n+1)^2

(n+1)[n(2n+1)/6 + (n+1)]
...here is where I'm lost, I'm not sure what I'm trying to manipulate it to look like...

Thanks!
 
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  • #2
mcraze123 said:
(n+1)[n(2n+1)/6 + (n+1)]
...here is where I'm lost, I'm not sure what I'm trying to manipulate it to look like...

Factor out 1/6 and then simplify what is left in the brackets.
 
  • #3
mcraze123 said:
induction step:
(0^2+1^2+2^2+...+n^2) + (n+1)^2 = (n+1)((n+1)+1)(2(n+1)+1)/6

n(n+1)(2n+1)/6 + (n+1)^2

(n+1)[n(2n+1)/6 + (n+1)]
...here is where I'm lost, I'm not sure what I'm trying to manipulate it to look like...
You're trying to show the first equation is true. To do this, you start with one side, as you have done, and manipulate it until it looks like the other side.
 

What is mathematical induction?

Mathematical induction is a proof technique used to prove statements about natural numbers or other well-ordered sets. It is based on the principle that if a statement is true for the first or base case and if it can be shown that the statement is also true for the next case, then it must be true for all subsequent cases.

How do you use mathematical induction to prove a statement?

To use mathematical induction to prove a statement, you first need to show that it is true for the base case, which is typically the smallest or simplest case. Then, you assume that the statement is true for the next case. Using this assumption, you must prove that the statement is also true for the next case. If you can do this, then the statement is true for all subsequent cases and the proof by mathematical induction is complete.

What is the difference between weak and strong induction?

In weak induction, you only use the previous case to prove the next case. In strong induction, you use all previous cases to prove the next case. This means that strong induction is a more powerful proof technique as it allows for a wider range of possible proofs.

When should I use mathematical induction?

Mathematical induction is typically used to prove statements about natural numbers or other well-ordered sets. It is particularly useful when the statement you are trying to prove involves a recursive definition or can be broken down into smaller cases. However, there are other proof techniques that may be more appropriate in certain situations.

What are some common mistakes to avoid when using mathematical induction?

One common mistake is assuming that the statement is true for all cases without actually proving it for each case. Another mistake is using the wrong base case or not considering all possible cases. It is important to carefully follow the steps of mathematical induction and ensure that each step is logically sound.

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