Which of these sequences whose nth term is given below is increasing?

Question

Which of these sequences whose nth term is given below is increasing? (4 points)

A
B
C
D

Answer 1

It's increasing when for all natural numbers n, a_n < a_n+1.

Let's look at choice A. a_n = 1/ n and a_n+1 = 1/(n+1). Therefore a_n+1 < a_n for all n, so the sequence is not increasing.

For choice B, we have a_n = n / (n+1) = 1 - 1/(n+1) and a_n+1 = (n+1) / (n+1+1) = (n+1)/(n+2) = 1 - 1/(n+2)

n+1 < n+2

Therefore, 1/(n+2) < 1/(n+1)

Therefore, -1/(n+2) > -1/(n+1) or -1/(n+1) < -1/(n+2)

Therefore, 1 - 1/(n+1) < 1 - 1/(n+2)

Therefore, a_n < a_n+1, so the sequence is increasing.

Thus, the answer is choice B.

Answer 2

Sequence with general term a_n = n/(n + 1) = 1 - 1/(n + 1) encreasing, because a_{n + 1} = 1 - 1/(n + 2) > 1 -

1/(n + 1) = a_n