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Proof by induction sigma 9^n-2^n is divisible by 7

Good day students welcome to put number nine on our review series of mathematical induction remember a collection of this coming on the mathgotserved.com on the precalc so before we take a look at that problem in this review installment let write down your plan of attack against the write of the plan that we're going to use to carry out the proof by induction sogonna break it into three steps step number one we have a base case okay so base case for the in this case what was going to do so going to show that the statement is true for the initial condition where the set of numbers that is likely that his and his problem we have the set of natural numbers so the smallest of the numbers one to for base cases that a show that n equals one true it is not always start one component the 92 three whatever Botsford in the case this problem were doing the whole set of natural numbers we start with one of Number two behalf the inductive hypothesis inductive hypothesis of the second part of our proof for inductive hypotheses are there to make an assumption within assumed that the missing that 'n equals K is true and it will case true for somek in the set of natural numbers and cases in n equals k is true for some K in a set of natural numbers and then the back portion into the longest is the most complicated is the inductive step inductive steps of my inductive step you have to show that show that and people K true by assumption implies that the next step and equals k+1 one is also true either this is a plan of attack right here is that what he used to two going into the act that would take a look at from the problem ninth were to prove that prove that to the n minus to to the is divisible by by seven for all n in the set of natural numbers okay now another way of phrasing this question is you have to prove that 7598-2 to the significant subject of the English state and not to do my proof i want right is the mathematical how do you express divisibility using equation right to how I could be like defining existing a mathematical formula right as a and the statement that night to the n minus 2 to the n indivisible 17 Camry to die 90 and -2 to the and equals seven and four some and in the set of natural numbers okay so there has indicated he can write any number at seven something integer then that number this by 70 to 95 both sides by that when you deliver the image on the right site again like in his effect altered from that number I so this is a statement that was going to be using to try to prefer right to list out part one part one remember the base case from a plan for base case slightly good to hear again's progress for fresh in my memory of these cases in a show at and equals one strict okay what is the second one because the set of natural numbers the first installment number is one now probably sure that while that easily take this statement right here in the plug-in one so we can take a look at exit 1 exit quietly to the nine to the first power -2 to the first hour and the response it is for thought is my -2 which is equal to seven and can write 77×1 which is equal to seven and four and equals one so the editor to satisfy this equally right here is the value and is equal to one a Texas 7001 indivisible by seven because you can write even informed consent integer indicates and equals one so what this show wishes that our base case is solid now on number two within a look at the inductive hypothesis okay inductive hypothesis is that they make an assumption here inductive hypothesis fighting inductive hypothesis that going to sell assume that and equals K is true I somehow we make that assumption that take a statement and with Dennis make NK and a semester I say let's go ahead and do that so it cannot think this statement is okay that's okay with me that in a hat denying to the K -22 okay is equal to seven and within assumed that this statement is true for some okay and and and a set of natural numbers so this is our assumption right here and what this statement basically means is that Thomas. The market's needs that

okay -230 K is what is invisible by seven something to forget that's what this statement I just kidding) part three is our inductive step inductive step for inductive step were going to show that the show that and equals K is true by assumption implies that the next step in equals cables one is also true indication of this then we can conclude by induction that the statement is in fact correct or the statement is always true right so let's go ahead of the start with S sub cables one at the next step after S okay and that the cables one is going to be denying it into and was honorably and keep us one -2 to keep us what now that make use of the properties of exp

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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