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Kingyou123
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Homework Statement
Homework Equations
Prof. Note's.
The Attempt at a Solution
I'm on the 3 line where my Prof. combines both equations, I'm confused on what my equation should look. Her's was (n+1)(n+1)+1)/2
You can write 5n+1 as 5 * 5n, and 5n+1 - 1 = 5 * 5n -5 +4 = 5(5n-1) + 4.Kingyou123 said:Homework Statement
View attachment 95427
Homework Equations
Prof. Note's. View attachment 95431
The Attempt at a Solution
I'm on the 3 line where my Prof. combines both equations, I'm confused on what my equation should look. Her's was (n+1)(n+1)+1)/2
View attachment 95428
Sorry, I just noticed this but should my equation be 4 l 5^n-1 or is what have okay?ehild said:You can write 5n+1 as 5 * 5n, and 5n+1 - 1 = 5 * 5n -5 +4 = 5(5n-1) + 4.
ehild said:Yes. As 5n-1 is divisible by 4 , the first therm of 5(5n-1) + 4 is divisible by 4, and the second term is just 4.
Sorry, I just refreshed my page and your comment appeared, thank you for the help :)HallsofIvy said:"4 l 5^n-1" is NOT even an equation!
HallsofIvy said:Saying that "4 divides [itex]5^n- 1[/itex] is NOT just a "reference" to [itex]\frac{5^n- 1}{4}[/itex]! It is the statement that [itex]\frac{5^n- 1}{4}[/itex] is an integer. That is, [itex]\frac{5^n- 1}{4}= k[/itex] for some integer k. Further the "n+1" form of the formula is not [itex]\frac{5^{n+1}- 1}{4}+ (n+1)[/itex]. I don't where you got that additional "(n+1)"! Replacing n by n+1 in [itex]\frac{5^{n}- 1}{4}[/itex] is just [itex]\frac{5^{n+1}- 1}{4}[/itex].
Now, of course, you want to "algebraically" go back to the "[itex]5^n[/itex]" and to do that use the fact that [itex]5^{n+1}= 5(5^n)[/itex].
It will be helpful to use [itex]5(5^n)- 1= 5(5^n)- 5+ 4[/itex].
I think that you have to go from the fact that if (5^n)-1 is divisible by 4 then (5^n)-1=4k where k is a constant. Now, how can you apply this to 5((5^n)-1)+4?. Think of substitutionKingyou123 said:Homework Statement
View attachment 95427
Homework Equations
Prof. Note's. View attachment 95431
The Attempt at a Solution
I'm on the 3 line where my Prof. combines both equations, I'm confused on what my equation should look. Her's was (n+1)(n+1)+1)/2
View attachment 95428
Induction proof is a mathematical method used to prove a statement or hypothesis for all cases by starting with a base case and then showing that if the statement is true for one case, it is also true for the next case. In science, induction proof is often used to show that a theory or principle holds true for all observations or experiments.
In strong induction proof, the statement being proved for the next case depends on multiple previous cases. In weak induction proof, the statement only depends on the immediately previous case. Strong induction proof is often used when the statement being proved is more complex and requires multiple previous cases to be true.
No, induction proof can only be used to prove statements that follow a specific pattern or structure. It cannot be used to prove statements that do not have a clear base case or do not follow a predictable pattern.
The base case is the starting point for an induction proof. It is the case for which the statement being proved is directly shown to be true. The base case is essential in induction proof because without it, there is no starting point to show that the statement holds true for all cases.
Yes, induction proof can be used in other fields such as philosophy, logic, and computer science. It is a logical and systematic method of proving statements and can be applied to any field that requires rigorous and logical reasoning.