Suppose that a sequence is defined by a1 1 an1 16an10 To s

Suppose that a sequence is defined by a_1 = 1, a_n+1 = 1/6(a_n+10). To show that a_n, is monotonic using mathematical induction, which of the following would be the second step? Assume 1/6(a_k+1 + 10) le 1/6(a_k + 10) and show that a_k+1 le a_k Assume a_k+1 le a_k and show that 1/6(a_k+1 + 10) le 1/6(a_k + 10) Assume 1/6(a_k+1 + 10) le 1/6(a_k + 10) and show that a_k+1 le a_k Assume a_k le a_k+1 ans show that 1/6(a_k+1 + 10) le 1/6(a_k + 10) Assume 1/6(a_k+1 + 10) le 1/6(a_k + 10) and show that a_k le a_k+1 Assume a_k le a_k+1 and show that 1/6(a_k+1 + 10) le 1/6(a_k + 10) Assume 1/6(a_k+1 + 10) le 1/6(a_k + 10) and show that a_k le a_k+1 Assume a_k+1 le a+k and show that 1/6(a_k+1 + 10) le 1/6(a_k + 10)

Solution

The number an is called the general term of the sequence {an} (nth term, especially for k = 1). The set {an : n k} is called the range of the sequence {an}nk. Sequences most often begin with n = 0 or n = 1, in which case the sequence is a function whose domain is the set of nonnegative integers (respectively positive integers). Simple examples of sequences are the sequences of positive integers, i.e., the sequence {an} for which an = n for n 1, {1/n}, {(1)n}, {(1)n + 1/n}, and the constant sequences for which an = c for all n. The Fibonacci sequence is given by a0, a1 = 1, a2 = 2, an = an1 + an2 for n 3. The terms of this Fibonacci sequence are called Fibonacci numbers, and the first few terms are 1, 1, 2, 3, 5, 8, 13, 21