Sums of Squares: Methods for Proving Identity Families

Abstract: This paper presents both a method and a result. The result presents a closed formula for the sum of the first $m+1,m \ge 0,$ squares of the sequence $F^{(k)}$ where each member is the sum of the previous $k$ members and with initial conditions of $k-1$ zeroes followed by a 1. The generalized result includes the known result of sums of squares of the Fibonacci numbers and a recent result of Schumaker on sums of squares of Tribonacci numbers. To prove the identities uniformly for all $k,$ the Algebraic Verification method is presented which reduces proof of an identity to verification of the equality of finitely many pairs of finite-degree polynomials, possibly in several variables. Several other papers proving families of identities are examined, and it is suggested that the collection of the uniform proof methods used in these papers could produce a new trend in stating and proving identities.