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?

   Hint (a):

   What is the pattern among the numbers he is adding?

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

   Hint (b):

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

   More Help (b):

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

   More Help (2b):

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