find three consecutive integers such that three times the sum of all three equals the product of the larger two

Question

this is my first time doing these type of problems and i have no idea where to start and where to finish. could you please help

Answer 1

The first integer is X

The second integer is X+1

The third integer is X+2

 

Three times the sum of all three....

 

3( X + X+1 + X+2) = 3( 3X + 3) = 9X + 9

 

 

the product of the larger two....

 (X+1)(X+2) = X^2 + 3x + 2

 

The problem says those expressions are equal...

 

9X+9 = X^2 + 3X + 2

 

0 = X^2 + 3X + 2 - 9X - 9

 

0 = X^2 - 6X - 7

 

0 =  (   X   -   7)(X   +    1)

 

X-7 = 0 --->  X = 7;

So 7, 8, and 9

 

3( 7 + 8 +9) = 3(24) = 72 = 8 * 9 <--- yes it works!!

 

X+1=0 ---> x = -1;

So -1, 0. and 1

 

3( -1 + 0 +1) = 3(0) = 0 = 0*1 <--- yes it works!!!

 

So there are 2 sets of consecutive integers that satisfy these conditions:

{7,8,9} and {-1,0,1}