Question

Determine whether the sequences are increasing, decreasing, or not monotonic. If increasing, enter 1 as your answer. If decreasing, enter −1as your answer. If not monotonic, enter 0 as your answer.
1. an=1/5n+7

2. an=n-5/n+5

3. an=(√n+5)/7n+5

4. an=cos(n)/5^n

Answer 1

Answer:

1) -1

2) 1

3) -1

4) 0

Step-by-step explanation:

1) This sequence is decreasing. The higher the value of n, the higher the value of the linear expression 5n+7. Since that expression is dividing 1, then the value of an will be smaller as long as n keeps growing. -1

2) This squence is increasing. For the first 5 values we have

a1 = -4/6 = -2/3

a2 = -3/7

a3 = -2/8 = -1/4

a4 = -1/9

a5 = 0

Which is increasing. Then the sequence takes positive values for the remaining values of n.

If n is higher than 5, then

an+1 = n-4/ n+6, and note that$an < an+1 \Leftrightarrow \frac{n-5}{n+5} < \frac{n-4}{n+6} \Leftrightarrow (n-5)*(n+6) < (n+5)* (n-4) \Leftrightarrow n^2+n-30 < n^2+n -20 \Leftrightarrow -30 < -20$

Which is true. As a cconsequence, the answer is 1

c) This sequence is decreasing. The polynomial that is in the denominator has higher degree and has higher ordered coefficients (and all of the positive). Thus, the higher the value of n, the higer the proportional difference between √n+5 and 7n+5 is. The answer is -1

4) This sequence is not monotonic. The positiveness of cos(n) might change for different values of n, for example

cos(1) > 0, then a1 >0

cos(2) < 0, then a2 <0

cos(3) <0

cos(4) <0

cos(5) >0, then a5 >0

As a consequence, a1 > a2 < a5. The sequence cannot be increasing nor decreasing, therefore the answer is 0.