# Patterns and variables

For a given pattern (1,4,9,16..)

What is the value for the nth number in the series and what is the pattern?

We have a difference in opinion with my son's 5th grade math teacher and want to get consensus.

• Just out of curiosity, what is the difference of opinion? This seems obviously to be the list of perfect squares. – lulu Dec 2 '17 at 14:32
• The teacher was using the recursive way referenced below to represent the sequence and given that we were struggling to get to the nth number. Using the list of perfect squares the answer was obvious. – Math Monkey Dec 2 '17 at 14:57
• Ah. Of course, those two expressions describe the same sequence. Worth noting that not all recursively defined sequences have pleasant closed formulas, so it is worth seeing both approaches. – lulu Dec 2 '17 at 15:22

## 2 Answers

This is the following succession: $$a_n=n^2$$.

$$a_1=1$$

$$a_n=a_{n-1}+(2n-1)$$

is a recursive way of representing the sequence of squares.

• @GuyFsone This might be one of the representations that have caused a 'difference in opinion' as stated by the OP. Adding consecutive odd numbers just might be the other way of writing sequence of squares that led to this. – Maadhav Gupta Dec 2 '17 at 15:15
• @GuyFsone My answer helped. Check out the comments on the OP. Not a low quality answer. – Maadhav Gupta Dec 2 '17 at 17:05
• obviously the answer is n square. can you prove that? – Guy Fsone Dec 2 '17 at 19:33
• ok I see what you mean +1 – Guy Fsone Dec 2 '17 at 19:34