Solution:All the triangles in this problem have base of length 2, and by the area formula, will have areas equal to their altitude. The area of the gold triangles: $$ 2+4+6+8+10+12+...+998 \\ = 2(1+2+3+4+5+6+...+499) \\ = 2\sum_{n=1}^{499} n \\ = 2 (499)(500)/2 \\ = 499 \cdot 500 \\ $$ The area of the black triangles: $$ 1+3+5+7+9+11+...+999 \\ = \sum_{n=1}^{500} 2n-1 \\ = 2 \sum_{n=1}^{500} n - \sum_{n=1}^{500} 1 \\ = 2(500)(501)/2 - 500 \\ = 500 \cdot 501 - 500 \\ = 500^2 \\ $$ The ratio of gold to black: $$ \dfrac{499(500)}{500(500)} = \dfrac{499}{500} $$ |