Let a[0 . . n] be an array of n + 1 natural numbers not exceeding n. let k < n be an integer such that the values of any two successive entries of a differ at most by k, i. e., |a[j] − a[j + 1]| ≤ k for all j ∈ {0, . . , n − 1}. 1. prove that there exist an index j such that |a[j] − j| ≤ (k + 1)/2. 2. given the number k, find an o(log n) divide and conquer algorithm that finds such an index.

Agroup of randomly selected apple valley high school students were asked to pick their favorite gym class. the table below shows the results of the survey. there are 528 students at apple valley high school. gym class number of students racquet sports 1 team sports 9 track and field 17 bowling 13 based on the data, what is the most reasonable estimate for the number of students at apple valley high school whose favorite gym class is bowling? choose 1 answer a. 9 b. 13 c. 119 d. 172