Alternatively, let the first term be $ A $, common difference $ d $, so terms: $ A, A+d, A+2d, A+3d $. Sum of squares of first and last:

["Understanding the Sum of Squares of First and Last Terms in an Arithmetic Sequence", "When analyzing arithmetic sequences, a common mathematical expression involves computing the sum of the squares of the first and last terms. Suppose we consider an arithmetic sequence defined by its first term ( A ) and common difference ( d ). The sequence progresses as follows:", "[\nA,\ A+d,\ A+2d,\ A+3d\ \ldots\n]", "For a sequence starting with the first four terms:\n[\n\ ext{First term: } A,\quad \ ext{Last term: } A + 3d\n]", "The problem asks for the expression representing the sum of the squares of these two terms:", "[\nA^2 + (A + 3d)^2\n]", "Expanding the square:", "[\nA^2 + (A^2 + 6Ad + 9d^2) = 2A^2 + 6Ad + 9d^2\n]", "This result highlights a concise algebraic expression for the sum of squares, valuable in sequences, optimization problems, and quadratic modeling.", "For a general arithmetic sequence starting with term ( A ) and common difference ( d ), the sum of the squares of the first and last terms (with ( n-1 ) terms total) remains ( A^2 + (A + (n-1)d)^2 ), which simplifies to", "[\n2A^2 + 2(n-1)Ad + (n-1)^2d^2\n]", "But in the specific case of four terms, where the last term is clearly ( A + 3d ), the expression simplifies neatly to:", "[\n\boxed{2A^2 + 6Ad + 9d^2}\n]", "This formula efficiently captures the information and supports further mathematical analysis, whether in algebra, calculus, or applied statistics involving sequence behavior.", "Keywords: arithmetic sequence, sum of squares, first term ( A ), common difference ( d ), sequential terms, quadratic expression, mathematical formula."]