### Home > INT3 > Chapter 10 > Lesson 10.2.1 > Problem10-109

10-109.

Julian’s math teacher, Ms. Pepperdine, has asked all of her students to memorize all of the perfect squares up to $20^{2}$. Julian hates memorizing, so he is trying to identify a pattern that will make it easier.

1. He lists the first five perfect squares: $1, 4, 9, 16, 25$ and looks for a pattern. “Hey, I see one! I add $3$, then $5$, then $7$, then $9$.” What does Julian mean? How can he find the next perfect square using his pattern?

What is the pattern among the numbers he is adding?

2. How can Julian prove that his pattern will work for all perfect squares?

If $2n$ must always result in an even integer,
what type of number must $2n + 1$ always equal?

What equation would show give the next consecutive odd number in the series?

Combine the equation $2n + 1$ with your equation from the Help before.
Simplify to find the proof that fits for all perfect squares.