Sum of the First n Positive Odd Numbers: A Proof Without Words

① This GeoGebra example demonstrates the classic identity that the sum of the first n positive odd numbers (1+3+5+...+(2n-1)) equals n², using a dynamic visual proof without words. ② The core mathematical concepts include arithmetic series, square numbers, and the integration of geometry and algebra, showing how odd numbers can be arranged to form a perfect square. ③ Construction highlights: using GeoGebra commands such as Sequence to generate the odd-number sequences, Intersect to create key points, and Text to label values; objects include a slider n, lists list1/list2/list3, point A, segments L_1/L_2, and multiple text labels. The visual arranges odd-numbered blocks into an n×n square, making the sum equal to the area of the square. ④ Suitable for middle school and above, this example is ideal for teaching series summation, algebraic identities, or mathematical proof, and serves as an engaging exploratory tool for students to connect visual patterns with symbolic formulas.