Multiple sums involving the terms of a general second order sequence of numbers

Closed forms are derived for nested finite sums of the form \[\sum_{a_{n-1}=c}^{a_n}\sum_{a_{n-2}=c}^{a_{n-1}}\cdots\sum_{a_0=c}^{a_1}x^{a_0},\] where \(a_n\) and \(c\) are integers and \(x\) is real or complex. This elementary identity is then used to evaluate multiple sums whose summands contain terms of the Horadam sequence \(\bigl(W_j(a,b;p,q)\bigr)\). The sequence is defined by \[W_0=a,\qquad W_1=b,\qquad W_j=pW_{j-1}-qW_{j-2}\quad(j\geq 2),\] where \(a,b,p,q\in\mathbb C\) with \(p\ne0\) and \(q\ne0\). The resulting identities include weighted sums involving Lucas sequences of the first and second kinds, Fibonacci and Lucas numbers, gibonacci sequences, and products of two and three shifted terms. The formulas show how the depth of summation is absorbed into binomial coefficients and shifted sequence indices, yielding compact expressions suitable for direct use in recurrence and summation problems.