It is convenient to do part b first, then use that result to do part a.
b. For some pair of points (x1, y1) and (x2, y2), you want to find a point (a, b) such that (a, b) - (x1, y1) = (x2, y2) - (a, b). That is, the differences of coordinates from one end to the center are the same as the differences from the center to the other end. Adding (a, b) to the above equation gives 2(a, b) - (x1, y1) = (x2, y2) Adding (x1, y1) then gives 2(a, b) = (x1, y1) + (x2, y2) Finally, dividing by 2 gives a formula for (a, b):
(a, b) = ((x1, y1) + (x2, y2))/2 The midpoint is the average of the end points.
a. Using the result from part b, the midpoint is midpoint = ((2, 3) + (6, 7))/2 = (8, 10)/2 midpoint = (4, 5)
The general formula for an arithmetic sequence is: <span>a(n) = a + d(n - 1) </span>
<span>a : first term </span> <span>d : common difference (6) </span> <span>n : number of the term </span>
<span>In your example, you are told the 6th term is 22. </span> <span>a(6) = a + 6(6 - 1) = 22 </span> <span>a + 30 = 22 </span> <span>a = 22 - 30 </span> <span>a = -8 </span>
<span>So now you have the general formula: </span> <span>a(n) = -8 + 6(n - 1) </span>
<span>If you like you can simplify it: </span> <span>a(n) = -8 + 6n - 6 </span> <span>a(n) = 6n - 14 </span>
<span>Then you can plug in n = 50: </span> <span>a(50) = 6(50) - 14 </span> <span>= 300 - 14 </span> <span>= 286</span>
