Question

Prove algebraically that the difference between the squares of any two consecutive integers is equal to the sum of these two integers.

Answer 1

Let any integer be represented by x. 
Then, the consecutive integer (the number that follows) should be x+1. 

The statement wants us to prove; 

(x+1)²-x²=x+(x+1) <-- solve left hand side 


x²+x+x+1²-x²
2x+1 (solution for left hand side) 

Now solve for right hand side. 

x+(x+1)= 2x+1 

As noticed, the LHS=RHS (left hand side= right hand side), therefore, the difference of squared consecutive numbers subtracted is equal to the sum of the two integers. 

Hope I helped :)