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DISCRETE MATHEMATICS

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Proof. If , we have sầu

on the left, and on the right we have

. Thus, the formula is true for .Assume then that the formula is true for some

. Then,

Thus, if the formula is true for

then it is true for

. Since we have sầu established that it is true for , we then have sầu that it is true for all

Update: From a request in the comments, we’ll add in a way lớn arrive at the formula (without just guessing).

First, we write,

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Then, we consider the product

Where in the last line we cancelled terms again. The only things we are left with are the

in the numerator & the 2 và

in the denominator. Of course, this is pretty much a proof that the formula is correct without using induction, but it doesn’t rely on us guessing the formula correctly.

As noted in the comments, often it is easier lớn guess the correct formula và use induction lớn prove the formula is correct than khổng lồ derive the formula directly.

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10 comments

January 10, 2021 at 1:09 am
Anonymous says:

alternatively, you can say that 1-1/4 = 1-(1/2) * 1+(1/2)= 1/2*3/2, and 1-1/9 = 1-(1/3)*(1+(1/3))= 2/3 * 4/3… then 3/2 * 2/3 = 1, and continue lượt thích that with the rest of the series, until finally all that is left is 1/2*(1+1/n) = 50% + 1/2/n = 1/2(1+1/n)= (1+n)/2n

March 13, 2019 at 9:51 am
Amber says:

I’m not quite understanding the rearranging step – could you please elaborate?

October 12, năm ngoái at 1:46 pm
Daniel Fugisawa says:

Amazing! Thank you again!

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Reply

October 12, 2015 at 12:46 pm
Daniel Fugisawa says:

Hi! Could you tell me how did you come up with the general law? Thanks

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