Height of the Book

Algebra Level 3

At the back of a bookstore, there is a storeroom where multiple copies of three types of books are kept. There are copies of a novel, an autobiography, and a linear algebra textbook. Each copy of the novel is 1cm thick, each copy of the autobiography is also 1cm thick while the linear algebra textbook is 2cm thick. Determine the number of ways to stack the books into a tower $n$ -cm high.

A. B. C. D. E.

E D A B C

1 solution

Raymond Julianto
Apr 29, 2020

Let $an$ be the number of ways to stack the books into a tower $n$ cm high. We consider the book at the bottom of the stack. If it is a novel, then there are $an-1$ ways to stack up the other books that will be $(n-1)$ cm high. If the bottom book is an autobiography, then again there will be $an-1$ ways to stack up the other books that will be $(n-1)$ cm high. If the bottom book is a linear algebra book then there are $an-2$ ways to stack up the other books that will be $(n - 2)$ cm high. So the recurrence relation is

You may wish to refer to this site for further explanation (different question, but similar)

https://medium.com/@andrew.chamberlain/the-linear-algebra-view-of-the-fibonacci-sequence-4e81f78935a3

The answer here (which is correct) is different from the option stated in the problem: $\frac{2 + \sqrt{2}}{4} ((1 + \sqrt{2})^n - (1 - \sqrt{2})^n).$

Jon Haussmann - 1 year, 1 month ago

Height of the Book

1 solution

1 pending report

Vote up reports you agree with