The Art and Craft of Problem Solving

160 CHAPTER 5 ALGEBRA

n n

and we can solve for}; p. We still need to sum the arithmetic series }; (3 j + I), but

J=! J=!

we already have a formula for this! Verify that

};

n.

}^2 - n(n+l)(2n+l)

J=!

• 6

Sometimes telescoping won't work with what you start with, but the introduction

of a single new term will instantly transform the problem. We call this the catalyst

tool. Once you see it, you will never forget it and will easily apply it to other problems.

Example 5.3.3 Simplify the product

Solution: Call the product P and consider what happens when we multiply P by

I - lla. The "catalyst" is the simple difference of two squares formula (x-y)(x+y) =

x^2 - i.

Hence

1-l1a^2101

P = ---'--,--

1 - lla

Infinite Series

Series with infinitely many terms is more properly a calculus topic, and you will find

more information in Chapter 9. For now, let us note a few elementary ideas. An infinite

series converges or diverges if its sum is finite or infinite, respectively. You may recall

the formula for the convergent infinite geometric series:

a+ar+ar +^2 ... = --, a

l- r