Induction Proof Help: Showing (n^5)/5 + (n^3)/3 + 7n/15 is Integer

Thread starter Anna Maria
Start date Oct 16, 2008
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Induction Proof

In summary, when beginning an induction proof, it is important to establish a base case and then use the inductive step to prove that the statement is true for all values of n. To show that a given expression is an integer, you can use mathematical induction by first proving the base case and then using the inductive step. Both the base case and the inductive step are necessary in an induction proof. However, induction can only be used to prove statements that follow a specific pattern. To write a clear and concise induction proof, it is important to clearly state the base case and inductive hypothesis, use organized steps and examples, and clearly state the conclusion and how it follows from the inductive step.

Oct 16, 2008

Anna Maria

I need to show that (n^5)/5 + (n^3)/3 + 7n/15 is an integer for all n.

I tried induction that obviously work for 1 but i could not manage to show this for k+1. Any tips please?

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Oct 16, 2008

rmcdra

try expanding the k+1 case. Pascals triangle is a huge help for figuring out how to expand it. After expanding and factoring, you should see the base case and your hypothesis.

1. How do I begin an induction proof?

To begin an induction proof, you first need to establish a base case. This is typically the smallest value of n for which the statement is true. Then, you assume the statement is true for some value k and use this assumption to prove that the statement is also true for k+1. This allows you to conclude that the statement is true for all values of n.

2. How do I show that (n^5)/5 + (n^3)/3 + 7n/15 is an integer?

To show that (n^5)/5 + (n^3)/3 + 7n/15 is an integer, you can use mathematical induction. First, establish the base case by plugging in the smallest value of n and showing that the result is an integer. Then, assume the statement is true for some value k and use this assumption to prove that the statement is also true for k+1. This will show that the statement is true for all values of n and therefore, (n^5)/5 + (n^3)/3 + 7n/15 is an integer.

3. Is it necessary to prove both the base case and the inductive step?

Yes, it is necessary to prove both the base case and the inductive step in an induction proof. The base case establishes the truth of the statement for the smallest value of n, while the inductive step proves that if the statement is true for some value k, then it is also true for k+1. Both steps are necessary to show that the statement is true for all values of n.

4. Can I use induction to prove any statement?

No, induction can only be used to prove statements that follow a specific pattern. The statement must be true for a base case and the inductive step must show that if the statement is true for some value k, then it is also true for k+1. If the statement does not follow this pattern, then induction cannot be used to prove it.

5. Are there any tips for writing a clear and concise induction proof?

Yes, there are a few tips for writing a clear and concise induction proof. First, clearly state the base case and the inductive hypothesis. Then, use clear and organized steps to prove the inductive step. It can also be helpful to use specific examples to illustrate the steps in your proof. Finally, make sure to clearly state your conclusion and how it follows from the inductive step.