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Proof by induction prove 1+5+9+13+ +(4n-3)=n(4n-2)/2
mathematical induction remember that a wide collection of this can be found on mathgotserved.com before we get started with so review the plan of attack plans that we going to use to guide our problem-solving process second part so lets write it down the plan to come out plan can be broken down into three parts for number one in the base case three cases the smallest value is set of natural numbers that can satisfy the statement okay so in this problem on your deal with the set of natural numbers so in this case for the base case were going to have to show after show that n equals one's true right that's the foundation of the base case step two is the inductive hypothesis right inductive hypothesis belittling an assumption here been assumed say that an equals case trip for some and a set of natural numbers and then third part longest and sometimes most complicated part of the Om proof is the inductive step okay inductive step sibling inductive step is to show that am n equals K is true follows that work like that n equals the next step keep us one is also true rights advocate complete a little bit can be done without proof I go ahead and take a look at the proof set of him for question number 10 when the proof that prove that one is five votes nine was 13+... +4 and -3 is equal to one over two and times four and -24 all n in a set of natural numbers That's going for set to get started lets start with part 1K so what are going to show for the base case for the base case step to show on that n equals one is true alright remembered a first natural number is one of the smallest one that there is an thing to do that we going to plugging one into both sides of the equation we have four and -3 is equal to one half and transport and -2 so plug-in wireless I will have 4×1-3? Is it equal to 1/2×1×4×1-2 is a true statement or not sort this out with employees for -31 is one equal to 1/2×4×1 is 4-2 is to just advice to the device further you have is one equal to one and the answer is yes so our base case is sound right let's move on to our home inductive hypothesis supported inductive hypotheses we're going to be making an assumption about assume assume that n equals K true for some K in the set of all natural numbers alright let's assume that that's true the what does that look like always going to do here is plug-in K for n in the original problem of Casey's's original problem right here we're going to plugging k for 'n both sides of our equation so that that look like this to the maker subject assume change, there on assume that that all the statement is one was five
+13+... Plus the 24 and -3
4K -3 is equal to one half times K times 4K -24 song K in the set of natural numbers are okay to Dennis R inductive hypothesis
in the inductive step further. The inductive step the hard part inductive step number going to do here for inductive step we want to show that it n equals K is true that follows that n equals the next step is keep us one is also true right to prove that the next it is true using this assumption of the important of your arguments are external links start by writing the left side of this statement of problem and it was limiting the next step case of right and left sides looking to have one spine.
+13+... Plus 4K -3 next step is lost for step four is bit of him K within a half people is one -3 I now is going to use our inductive hypotheses here to make a substitution rights alone substitution was going to make this piece of our inductive step right near begin substituting with another value that we stated with equal to our inductive hypothesis we know that this entire expression is equal to one half key times 4K -2 rights... On the best substitution here so this is that equal to the equal to one half K times 4K -2 advocate simplifies a little bit +4 times K okay 4×1 is 4-03 use one of one there edited the site appear by assumption in part 2O making his or assumption to I now select go ahead and that simplify this a little bit further selecting what can we do here well you noted to me come I will life easier is good to how idea what the target is okay; make note of what the target should look like it, simplify individual format easy to write back the page so what is our destination results going to be so while it's that look like is the right part of her inductive step with all Kay's replacement keep us what so what better look like we can have one half is the type of target is one half times into an output keep us one times for times cases 1-22 this is what I want this and expression to simplify into okay it's good to do this we can have idea how to manipulate you are expression to get your initial here intended target is anything but a little bit more so I can have all an easy expression to intellect
Tagged under: 5^-2^ divisible 3,^3+2n divisible 3,8^-1 divisible 7,1/(1X2)+1/(2X3)+...+1/((+1))=/(+1),^2 than2n,8+2x8+3x8+..+8n=,1+3+5+7+...+(2n-1)=^2 N,sum ^2= 1/6n(+1)(2n+1),9^-2^ id divisible 7,1+5+9+13+..+(4n-3)=½(4n-2), Σ =(+1)/2,1^3+2^3+3^3+..+^3= ((+1)/2)^2 ^2(+1)^2/4,10^1+10^2+10^3+ +10^ = 10/9(10^-1),derivative ^-1 ( (-1)^ !)/ ^(+1),2^ greater equal 1+,3^ thatn (+1)!
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