On Certain Relation of Sums of Squares and Quartic ∑ᵣ₌₁2 k aᵣ⁴+k d⁴=2 ∑ᵣ₌₁2 k-1(aᵣ aᵣ₊₁+d²)²

Let k and d be a given positive integers and suppose that ar is a given sequence. In this current study, we investiagate a diophantine identity relating the sums of squares and quartic from specific sequences to a variable d. In particular, the diophantine identity \(∑ᵣ₌₁2 k a_r^4+k d^4=2 ∑ᵣ₌₁2 k-1(a_r aᵣ₊₁+d^2)^2\) is developed and introduced. The objective of this research is to determine the conditions under which integer solutions for (ar,d) exist within this diophantine equation. The methodology involves, decomposing polynomials, factorizing polynomials, and exploring the solution set of the given equation.