Prove Even Integer for Natural Number n: Induction

In summary, to prove that \frac{n^{3}+5n}{3} is an even integer for each natural number n, use mathematical induction and replace n+1 in the equation. Divide each term by 3 and see that the result is a whole number. This should be enough if you are familiar with induction.
  • #1
asset101
11
0
Use mathematical induction, to prove that [tex]\frac{n^{3}+5n}{3}[/tex]


is an even integer for each natural number n.

I am fimilar with proof by induction but in most of the question that I have done have a
LHS = RHS which seems to simplifiy things a little bit.
Any help would be appreciated
Cheers
 
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  • #2
asset101 said:
Use mathematical induction, to prove that [tex]\frac{n^{3}+5n}{3}[/tex]
is an even integer for each natural number n.

I am fimilar with proof by induction...

Put n+1 in place of n.

(n+1)^3 + 5(n+1) = n^3+3n^2+3n+1+5n+5 = (n^3+5n) + 3n(n+1) + 6.

Now divide each term by 3 and see what kind of number you get.

Since you are familiar with induction, this should be enough.
 
  • #3
Got it thanks mate
 

1. What does it mean to prove an even integer for a natural number using induction?

Proving an even integer for a natural number using induction means to demonstrate that for any given natural number, the next consecutive natural number is also even. This is done by first proving the base case (usually n = 0 or 1) and then showing that if the statement holds true for n, it also holds true for n+1. This creates a chain of logic that proves the statement for all natural numbers.

2. Why is induction used to prove even integers for natural numbers?

Induction is used because it is a powerful mathematical tool that allows us to prove statements for all natural numbers without having to explicitly check each individual case. It relies on the assumption that if a statement is true for a given natural number, it is also true for the next consecutive natural number, and by repeating this logic, we can prove the statement for all natural numbers.

3. Can induction be used to prove other types of statements besides even integers for natural numbers?

Yes, induction can be used to prove a wide range of mathematical statements, including but not limited to divisibility, inequalities, and identities. It is a versatile technique that can be applied to many different types of problems in mathematics.

4. Are there any limitations to using induction to prove even integers for natural numbers?

One limitation of induction is that it can only be used to prove statements that are true for all natural numbers. If a statement is only true for a subset of natural numbers, then induction cannot be used to prove it. Additionally, the base case for induction must be explicitly stated and proved, which can sometimes be difficult or time-consuming.

5. Can induction be used to prove odd integers for natural numbers?

Yes, induction can be used to prove statements about odd integers for natural numbers. However, the proof would be slightly different as it would start with the base case of n = 1 (since 0 is an even number) and then show that if the statement holds true for n, it also holds true for n+2. This would create a chain of logic for all odd natural numbers.

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