NP Chapter 2 Page 62 Exercise 17

[ray_serrano]

I can see how we get to k2 = 4n^2 -4n + 1, using FOIL. But how do we factor to k = 4n(n-1)+1?

[twallack]

I thing that's an error in the book. The left side of the equation shouldn't change.

[RonPurewal]

ah yes, that should be k^2, not k.

we just pulled the 4n out of the first two terms. that's a bit of an ingenious transformation - it's probably not something i would think to do, for instance, and i teach the test!

i'll call attention to the error.

--

by the way, if you ever see something like this on the real test, you're probably better off not looking for this sort of mathe-magic.

instead, just do PATTERN RECOGNITION. plug in a bunch of numbers and look for a pattern.

if k = 2n - 1, then the first k's that come to mind are 1, 3, 5, 7, and 9.
k = 1 --> k^2 = 1, remainder 1
k = 3 --> k^2 = 9, remainder 1
k = 5 --> k^2 = 25, remainder 1
k = 7 --> k^2 = 49, remainder 1
k = 9 --> k^2 = 81, remainder 1

the pattern is clear.