Note that a product may be parenthesized in two different ways: and Similarly, there are several different ways to parenthesize Two such ways are and Two such ways are and Let be the number of different ways to parenthesize the product Show that if then for every integer (It turns out that the sequence is the same as the sequence of Catalan numbers: for every integer See example 5.6.4.)
To show that the different number of ways to parenthesize is for all integers , if .
There are different ways to parenthesize the product for e.g. can be parenthesized in two different ways and . be the different number of ways to parenthesize then for every integer and .
Since it is given that and also observe that and .Since the product of one or two variables can’t be parenthesized in more than one way. So it is logically correct.
Also the product can be parenthesized in two different ways and
Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!Get Started