Sequence and Series ?

Question

The fourth power of the common difference of an arithmetic progression with integer entries is added to the product of any four consecutive terms of it.
Prove that the resulting sum is the square of an integer.

1748 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-04-06T11:35:52

Refer to The Explanation.

As we need to consider #4# consecutive terms of an A.P., it will

be easier to take the

#4" terms "x-3y,x-y,x+y, and, x+3y.#

The common difference, then, is #2y.#

To prove that, #(x-3y)(x-y)(x+y)(x+3y)+(2y)^4# is a perfect suare.

#"The Exp.="(x-3y)(x-y)(x+y)(x+3y)+(2y)^4,#

#=[{(x-3y)(x+3y)}{(x-y)(x+y)}]+16y^4,#

#=[(x^2-9y^2)(x^2-y^2)]+16y^4,#

#=x^4-10x^2y^2+9y^4+16y^4,#

#=x^4-10x^2y^2+25y^4,#

#=(x^2-5y^2)^2.#

Hence, the Proof.