A subset T of the integers is defined recursively as follows: Base case: 2 ∈ T Recursive rule: if k ∈ T, then k + 5 ∈ T This problem asks you to prove that T is exactly the set of integers that can be expressed as 5m+2, where m is a non-negative integer. In other words, you will prove that x ∈ T if and only if x = 5m+2, for some non-negative integer m. The two directions of the "if and only if" are proven separately. (a) Use structural induction to prove that if k ∈ T, then k = 5m + 2, for some non-negative integer m.

Match the following terms with their definitions. 1. bisector of a segment ray ba and ray bc are opposite rays if a, b, and c are collinear and b (the endpoint of both rays) is between a and c. 2. opposite rays a ray, , is the set of points beginning at point a and going infinitely in the direction of point b. 3. collinear points a line or segment that intersects the segment at its midpoint. 4. betweenness of points a set of two or more points all on the same line. 5. ray the distance between the endpoints of a segment. 6. space point b is between a and c if a, b, and c are collinear and the equation ab + bc = ac is true, where ab, bc, and ac are the distances between points a and b, b and c, and a and c, respectively. 7. midpoint of a segment a set of two or more points all on the same plane. 8. coplanar points the point on a segment that divides the segment into two equal segments. 9. length of a segment the set of all possible points. 10. line segment the set of two different endpoints and all points between them.

Aculture started with 2,000 bacteria. after 8 hours, it grew to 2,400 bacteria. predict how many bacteria will be present after 19 hours . round your answer to the nearest whole number. p=ae^kt

Select all of the following expressions that are equal to 2.5.