Why is any number squared equal to the the number before it times the number after it plus one?

Question

Why is any number squared equal to the the number before it times the number after it plus one?

Basically:

x(x+2)+1=(x+1)^2

Does this work for all numbers??

In: Mathematics

Answer 1

Arrange a bunch of, say, coins in a square grid. If there are *n* rows and *n* columns of coins, there will be *n*^2 coins in total. Now remove one row of *n* coins and use these to add an extra column. But, because you now have one fewer row, you only need *n*−1 coins for the new column. You have one coin left over. This demonstration works for any whole number *n* from 2 up.