Proof. If , we have on the left, và 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 established that it is true for , we then have that it is true for all Update: From a request in the comments, we’ll địa chỉ in a way lớn arrive at the formula (without just guessing).

First, we write, " title="Rendered by QuickLaTeX.com"/>

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 và the 2 & 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 to guess the correct formula và use induction lớn prove the formula is correct than to lớn derive the formula directly.

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### Related

Find the error in an inductive “proof”
Establish a formula for the sản phẩm (1-1/2)(1-1/3)…(1 – 1/n) 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, và 1-1/9 = 1-(1/3)*(1+(1/3))= 2/3 * 4/3… then 3/2 * 2/3 = 1, & continue like that with the rest of the series, until finally all that is left is 1/2*(1+1/n) = 1/2 + 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, 2015 at 1:46 pm
Daniel Fugisawa says:

Amazing! Thank you again! October 12, năm ngoái 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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If you are having trouble with math proofs a great book lớn learn from is How to lớn Prove It by Daniel Velleman:  A really awesome book that I highly recommend on how lớn study math và be a math major is Laura Alcock"s, How khổng lồ Study as a Mathematics Major:  My Complete Math Book Recommendations:
Basics: Calculus, Linear Algebra, và Proof WritingCore Mathematics Subjects. 