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## New AOPS Precalculus Book!

Richard Rusczyk’s long awaited Precalculus Book is now available!  Here is the description:

Precalculus is part of the acclaimed Art of Problem Solving curriculum designed to challenge high-performing middle and high school students. Precalculus covers trigonometry, complex numbers, vectors, and matrices. It includes nearly 1000 problems, ranging from routine exercises to extremely challenging problems drawn from major mathematics competitions such as the American Invitational Mathematics Exam and the USA Mathematical Olympiad. Almost half of the problems have full, detailed solutions in the text, and the rest have full solutions in the accompanying Solutions Manual.

As with all of the books in Art of Problem Solving’s Introduction and Intermediate series, Precalculus is structured to inspire the reader to explore and develop new ideas. Each section starts with problems, so the student has a chance to solve them without help before proceeding. The text then includes solutions to these problems, through which new techniques are taught. Important facts and powerful problem solving approaches are highlighted throughout the text.

About the authors: Richard Rusczyk is the founder of http://www.artofproblemsolving.com. He is co-author of the Art of Problem Solving, Volumes 1 and 2 and Intermediate Algebra, and author of Introduction to Algebra and Introduction to Geometry. He was a national MATHCOUNTS participant, a three-time participant in the Math Olympiad Summer Program, a perfect scorer on the AIME, and a USA Math Olympiad Winner. The solutions are co-authored by Naoki Sato. He is a curriculum developer and the director of WOOT at Art of Problem Solving. He won first place in the 1993 Canadian Mathematical Olympiad and is a 2-time medalist at the International Mathematical Olympiad. He has been Deputy Leader of the Canadian IMO team three times.

ISBN: 978-1-934124-16-1
Text: 528 pages. Solutions: 272 pages.
Paperback. 10 7/8 x 8 3/8 x 1 1/16 inches.

## Telescoping Sums and Products (Example 1)

While we wait for the Metroplex Math Circle to continue again in mid January, I will be posting a series of problems and solutions from our last session. Those who were fortunate enough to attend learned useful techniques for solving problems common in math contests.

### Telescoping Sums and Products

The telescoping sums and products idea is used to solve many problems involving sums or products in algebra. For problems involving sums, the idea is to use identities, to write the sum in the form

$\displaystyle\sum^{n}_{k=1}[F(k+1)-F(k)]$

and then cancel out terms to get F(n+1)-F(1). Sometimes the desired identity is hard to find, but basically you are searching for it in the recursive form of the sequence, or you can look foor the “conjugates” for the terms you have. The first example is classical. You Certainly know these formulas.

$\displaystyle\sum^{n}_{k=1}k=\dfrac{n(n+1)}{2}$

$\displaystyle\sum^{n}_{k=1}k^2=\dfrac{n(n+1)(2n+1)}{6}$

$\displaystyle\sum^{n}_{k=1}k^3=\left[\dfrac{n(n+1)}{2}\right]^2$

What about $\displaystyle\sum^{n}_{k=1}k^4$?

Example 1.  Prove that

$\displaystyle\sum^{n}_{k=1}k^4=\dfrac{n(n+1)(2n+1)(3n^2+3n-1)}{30}$