Question

Use mathematical induction to prove the statement is true for all positive integers n, or show why it is false. 4 x 6 + 5 x 7 + 6 x 8 +...+ 4n(4n+2) = 4(4n+1)(8n+7)/6

Answer 1

First put it in the sum of a series formula.

Answer 2

\[\sum_{r=1}^{n}4n(4n+2)\]

Answer 3

Although that doesn't appear to be correct because that would give you 4x6 + 8x10 + 12x14......

Answer 4

@phi Any input?

Answer 5

what is the value for n=1 ?

Answer 6

I mean, what is the sum of a series formula here?

Answer 7

Oh, sorry just saw this. But if I plug in n =1, then I get 2

Answer 8

I am completely lost, so will you please walk me through it from step one?

Answer 9

I would have said it was\[\sum_{r=1}^{n}(r+3)(r+5)\]

Answer 10

for n=1 there is only one term in the summation
4n(4n+2) --> 4*6
which is 24

however
4(4n+1)(8n+7)/6 with n=1 is not 24

so we are not doing too well...

Answer 11

Well it says to show why it is false... So I just will say that it is false and therefore cannot be proved true.

Answer 12

4n(4n+2) is incorrect isn't it?? Surely it's (n+3)(n+5)?

Answer 13

4n(4n+2) is not 5x7 when n=2.

Answer 14

Or 6x8 when n=3.

Answer 15

Use mathematical induction to prove the statement is true for all positive integers n, or show why it is false.

In an induction proof, you start by proving a "base case"
in this problem, the base case (for n=1) is not true, so the formula is proven false at the first step.

Answer 16

Okay so now I plug in 1 everywhere were n is.. getting 50

Answer 17

So, I show my work for n=1, then state that it is false?

Answer 18

yes, and the series has only one term which clearly is 24
24 ≠ 50
therefore, false

Answer 19

Okay, thank you!

Answer 20

Are you just ignoring everything i'm saying?

Answer 21

Noo! I just didn't understand what you meant.

Answer 22

No, I meant phi :)

Answer 23

This series says that the last term is (4n)(4n+2). I'm saying that that can't be right because all of the previous terms were (n+3)(n+5). If n is 1 you get 4x6. If n is 2 you get 5x7. If n is 3 you get 6x7.
If you use 4n(4n+2), that doesn't work with the other terms.

Answer 24

I'm going to post a new question and close this one.. unless you two would like to help.

Answer 25

& OHHH,ok.

Answer 26

Because if n=2 then you get 8x10. If n=3 then you get 12x14.

Answer 27

@phi Is what I've said not correct?

Answer 28

@alrightatmaths probably, but it doesn't make the formula correct.