- #1
kathrynag
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Homework Statement
I need to do this by proof by induction
8n divides (4n)!
Homework Equations
The Attempt at a Solution
I already did it for 1, now I need to do for k and k+1
(4k)!t=8k
Induction is a mathematical proof technique that involves proving a statement for all integers starting from a base case and then using the fact that if the statement holds for a certain integer, it also holds for the next integer. In this particular proof, we will start with the base case of n=1 and then show that if the statement is true for n=1, it is also true for n+1.
The base case for this proof is when n=1. We will show that 8 divides 4, which is true because 4=8*0. Therefore, the statement holds for n=1.
To prove the induction step, we assume that the statement holds for n=k and then show that it also holds for n=k+1. In this proof, we will assume that 8k divides 4k and then use algebraic manipulation to show that 8(k+1) divides 4(k+1). This will prove that if the statement is true for n=k, it is also true for n=k+1.
No, this proof only works for positive integers. Additionally, the statement will only hold for values of n that are divisible by 4. For example, n=2 will not work because 8 does not divide 8.
This proof is important because it demonstrates the power of mathematical induction in proving statements for all integers. It also shows the relationship between divisibility and multiplication, which is a fundamental concept in mathematics. Induction is a widely used proof technique in many areas of mathematics and this proof serves as a good example of its application.