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Meehan98
 one year ago
Is anyone good at mathematical induction? I'm so confused!
Meehan98
 one year ago
Is anyone good at mathematical induction? I'm so confused!

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Meehan98
 one year ago
Best ResponseYou've already chosen the best response.0I know that the first step is to make n=1 in the equation, but I don't understand the "n=k+1" part. Do I just input "k+1" every time there's an "n" in the equation?

welshfella
 one year ago
Best ResponseYou've already chosen the best response.1Proof by induction first attempts to prove that the formula is true for k = 1. The process assumes that the result is true for n = k. Then the (k+1 )th term is added. If the resulting expression is the same as the expression for the sum for k terms except that k is replaced by (K+1) then this constitutes the proof.

welshfella
 one year ago
Best ResponseYou've already chosen the best response.1 because if iot is true for n = 1 then it must be true for n = 2 , 3 etc

Meehan98
 one year ago
Best ResponseYou've already chosen the best response.0So, the first step would give you 1+3+5+...+(1)=1

Meehan98
 one year ago
Best ResponseYou've already chosen the best response.0And then if n=k, you get: 1+3+5+...+(2k1)=k^2

welshfella
 one year ago
Best ResponseYou've already chosen the best response.1No the first step is what is the sum of n terms if n = 1? 1 = first term so sum of 1 term = 1 n^2 = 1^2 = 1 = sum of 1 term so true for n = 1

welshfella
 one year ago
Best ResponseYou've already chosen the best response.1yes that is the sum of k terms

welshfella
 one year ago
Best ResponseYou've already chosen the best response.1now add the (k + 1)th term

welshfella
 one year ago
Best ResponseYou've already chosen the best response.1what is the formula for (k+1)th term kth term = 2k  1 right?

Meehan98
 one year ago
Best ResponseYou've already chosen the best response.0Ok, is this looking right so far? 1+3+5+...(2k1)+(k+1)=k^2+(k+1)

Meehan98
 one year ago
Best ResponseYou've already chosen the best response.0Because I add the (k+1)th term to each side?

welshfella
 one year ago
Best ResponseYou've already chosen the best response.1no the (k+1)th term is (2(k + 1)  1) dw:1443022249395:dw

welshfella
 one year ago
Best ResponseYou've already chosen the best response.1the k is replaced by (k + 1)

Meehan98
 one year ago
Best ResponseYou've already chosen the best response.0Oh, okay that makes sense. Do we do the same thing to the other side of the equation?

Meehan98
 one year ago
Best ResponseYou've already chosen the best response.0If so, the equation I have so far is 1+3+5+...+(2k+1)1)=(k+1)^2

welshfella
 one year ago
Best ResponseYou've already chosen the best response.1yes the right side of the equation now becomes k^2 + 2(k + 1)  1 which if you expand comes to (k +1)^2

welshfella
 one year ago
Best ResponseYou've already chosen the best response.1 and thats your proof (k + 1)^2 is k^2 with the k replaced by (k +1)

Meehan98
 one year ago
Best ResponseYou've already chosen the best response.0Okay, would we leave (2k1) in the equation, because the first option looks like the correct answer but it still has (2k1) in it?

welshfella
 one year ago
Best ResponseYou've already chosen the best response.1yes leave 2k1 there the first option is the correct one

welshfella
 one year ago
Best ResponseYou've already chosen the best response.12k1 is the kth term you need to find the general formula for any term in order to do the proof

welshfella
 one year ago
Best ResponseYou've already chosen the best response.1its a bit tricky to learn Induction . Practice will be the best way to grasp it.

Meehan98
 one year ago
Best ResponseYou've already chosen the best response.0Thank you! Yes, it's a lot to take in at first. Do you have time to help me with one more?

welshfella
 one year ago
Best ResponseYou've already chosen the best response.1First plug n = 1 into the right hand formula and see if it gives 1

Meehan98
 one year ago
Best ResponseYou've already chosen the best response.0So, my first step is to make n=1 and I get 1^3=1(4)/4 and this is true.

welshfella
 one year ago
Best ResponseYou've already chosen the best response.1yea good work now let n = k and then write formula with k

Meehan98
 one year ago
Best ResponseYou've already chosen the best response.0My next step is to make n=k and I get: 1^3+2^3+3^3+...+k^3=k^2(k+1)^2/4

welshfella
 one year ago
Best ResponseYou've already chosen the best response.1now add (the (k + 1)th term to both sides

Meehan98
 one year ago
Best ResponseYou've already chosen the best response.0Now i replace k with (k+1) (k+1)^3=(k+1)(k+1)+1)^2/4

welshfella
 one year ago
Best ResponseYou've already chosen the best response.1No you ADD (k + 1)^3 to both sides You are aiming for the formula that you have written but havent got it yet

welshfella
 one year ago
Best ResponseYou've already chosen the best response.1RHS is k^2 + (k + 1)^2 + (k + a)^3  4

welshfella
 one year ago
Best ResponseYou've already chosen the best response.1you now have to simplify this

welshfella
 one year ago
Best ResponseYou've already chosen the best response.1sorry its not a plus there correct is k^2 (k + 1)^2  + (k + 1)^3 4

Meehan98
 one year ago
Best ResponseYou've already chosen the best response.0This is so confusing; I thought that you would multiply by 4 to each side to get rid of the denominator, but you don't. I don't know the first step in order to simplify this. Sorry.

welshfella
 one year ago
Best ResponseYou've already chosen the best response.1make this into one fraction

welshfella
 one year ago
Best ResponseYou've already chosen the best response.1no you multiply the (k + 1)^3 by 5 so its k^2(k + 1)^2 + 4(k + 1)^3  4

welshfella
 one year ago
Best ResponseYou've already chosen the best response.1now take out the common factor (k + 1)^2 = (k + 1)^2( k^2 + 4k + 4)  4

welshfella
 one year ago
Best ResponseYou've already chosen the best response.1* i should have said multiply by 4

welshfella
 one year ago
Best ResponseYou've already chosen the best response.1do you follow the above?

Meehan98
 one year ago
Best ResponseYou've already chosen the best response.0Yes, so you get: (k+1)^2 (k+2)^2/4

Meehan98
 one year ago
Best ResponseYou've already chosen the best response.0Then you get: (k+1)^2 ((k+1)+1)^2/4 which is the original n^2(n+1)^2/4

welshfella
 one year ago
Best ResponseYou've already chosen the best response.1exactly which is the formula for sum of k terms with the k replaced by k+1 So if the formula is trye for n = k then its also true for n = k + 1.

welshfella
 one year ago
Best ResponseYou've already chosen the best response.1we've also shown that its true for n = 1 so it must be true for n = 2,3 etc True for all whole number values of n.

Meehan98
 one year ago
Best ResponseYou've already chosen the best response.0Thank you! You've been wonderful help!

welshfella
 one year ago
Best ResponseYou've already chosen the best response.1for practice try a relatively easy one show by induction 1+2+3+ + n = (n/2)(n + 1)

Meehan98
 one year ago
Best ResponseYou've already chosen the best response.0Okay, I'm close but not there yet. After making the n=1 and n=k I got: k=k/2(k+1) now I add (k+1)th term and I got: (k+1)=k/2 (k+1)+(k+1), but that would equal (k+1)=k/2 2(k+1) so I'm doing something wrong.

welshfella
 one year ago
Best ResponseYou've already chosen the best response.1we have (k/2)(k + 1) + k + 1 take out the common factor (k + 1) = (k + 1( (k/2) + 1) = ( k + 1) ( k + 2)  2 = (k + 1) (( k + 1) + 1)  2 which is the formula for sum of k terms with k replaced by (k + 1)

welshfella
 one year ago
Best ResponseYou've already chosen the best response.1you have to 'juggle ' the terms about a bit  this is the trickiest part

welshfella
 one year ago
Best ResponseYou've already chosen the best response.1I thought this was straight forward but it wasnt!

welshfella
 one year ago
Best ResponseYou've already chosen the best response.1 a good idea is to write down what you are aiming for before you attempt the algebra

Meehan98
 one year ago
Best ResponseYou've already chosen the best response.0Okay, thank you for all of your help!
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