Mathematics Magazine for Grades 112 



Theory: The
Conic Sections. For any of the below with a center (j, k) instead of (0, 0),
replace each x term with (xj) and each y term with (yk).
1. Find the twentieth term and the sum of the first 20 terms of the arithmetic progression 4, 9, 14, 19…. Solution: For this progression a = 4, d = 5, and n = 20; The twentieth term is l = a + (n  1) ∙ d = 4 + 19 ∙ 5 = 99. And the sum of the first 20 terms is: = 1030 2. Insert five arithmetic means between 4 and 22. Solution: We have a = 4, l = 22, and n = 5 + 2 = 7. Then 22 = 4 + 6 ∙ d and d = 3. The first mean is 4 + 3 = 7, the second is 7 + 3 = 10, and so on. The required means are: 7, 10, 13, 16, 19 and the resulting progression is 4, 7, 10, 13, 16 19, 22 3. Find the arithmetic mean of the two numbers a and l. Solution: We seek the middle term of an arithmetic progression of three terms having a and l as first and third terms, respectively. If d is the common difference, then a + d = l – d and d = (l  a). The arithmetic mean is a + d = a + (l  a) = (l + a). 4. Find the ninth term and the sum of the first mine terms of the geometric progression 8, 4, 2, 1,…. Solution: Here a = 8, and r = ½, and n = 9; The ninth term is l = ar^{n1}= The sum of the first nine terms is:
Proposed Problems: 1. Graph each function and state its domain and range. y = 3x^{2} + 4 2. For the following parabola find: i) the direction of opening y = x^{2} + 3 3. Find the equation of each parabola vertex at (0, 2) and passing through the point (3,7) 


