SOLUTION: Sorry! I put thwe wrong email address. The correct one is PayJ3@aol.com "Prove the product of 4 consecutive integers plus 1 is the square of an odd number (always)"

Algebra ->  Problems-with-consecutive-odd-even-integers -> SOLUTION: Sorry! I put thwe wrong email address. The correct one is PayJ3@aol.com "Prove the product of 4 consecutive integers plus 1 is the square of an odd number (always)"      Log On


   

Question 89693: Sorry! I put thwe wrong email address. The correct one is PayJ3@aol.com

"Prove the product of 4 consecutive integers plus 1 is the square of an odd number (always)"
Answer by scott8148(6628) About Me  (Show Source):
You can put this solution on YOUR website!
let x equal the first integer ... x(x+1)(x+2)(x+3)+1=x^4+6x^3+11x^2+6x+1

since this is a perfect square and the coefficients of the first and last terms are 1, the factors must be of the form x^2+bx+1

(x^2+bx+1)^2=x^4+2bx^3+(b^2+2)x^2+2bx+1 ... so b=3 ... x^2+3x+1=x(x+3)+1

integers that are separated by an odd amount (3) must be odd and even; so their product must be even ... adding 1 makes it odd

ALWAYS