Which of the following statements is true for the sequence with general term a sub n equals the quotient of the product of the quantities n plus 2 and the quantity n plus 4 and n raised to the 4th power , for n = 1, 2, 3…? (10 points)
A) It is decreasing.
B) It is not monotonic.
C) It is divergent.
D) It is bounded above but not bounded below.
use the ratio test. If a_(n+1)/a_n < 1 it converges, since all the terms are positive. Also review the ratio test.
To determine which of the given statements is true for the sequence with the general term a sub n, we need to analyze the properties of the sequence.
The general term of the sequence is given as a sub n equals the quotient of the product of the quantities n plus 2, n plus 4, and n raised to the 4th power. Let's write it mathematically: a sub n = (n + 2)(n + 4)/(n^4).
Now, let's evaluate the first few terms of the sequence to gain insights:
For n = 1:
a sub 1 = (1 + 2)(1 + 4)/(1^4) = (3)(5)/(1) = 15
For n = 2:
a sub 2 = (2 + 2)(2 + 4)/(2^4) = (4)(6)/(16) = 24/16 = 3/2
For n = 3:
a sub 3 = (3 + 2)(3 + 4)/(3^4) = (5)(7)/(81) = 35/81
Based on the first few terms of the sequence, it is not immediately clear whether it is decreasing, monotonic, divergent, or bounded above/below. Thus, we need to evaluate further terms of the sequence to make a determination.
For n = 4:
a sub 4 = (4 + 2)(4 + 4)/(4^4) = (6)(8)/(256) = 48/256 = 3/16
For n = 5:
a sub 5 = (5 + 2)(5 + 4)/(5^4) = (7)(9)/(625) = 63/625
From the evaluation of these terms, we can observe that the sequence is not monotonic, as it does not strictly increase or decrease throughout. However, we need to analyze it further to determine if it is increasing or decreasing on average.
Now, let's calculate the ratios of consecutive terms of the sequence:
a sub 2 / a sub 1 = (3/2)/(15) = 1/10
a sub 3 / a sub 2 = (35/81)/(3/2) = 70/81
a sub 4 / a sub 3 = (3/16)/(35/81) = 243/560
a sub 5 / a sub 4 = (63/625)/(3/16) = 336/625
From these ratios, it can be observed that the sequence is decreasing on average. Thus, we can conclude that statement A) "It is decreasing" is true for the given sequence.
To summarize, the correct statement for the given sequence is:
A) It is decreasing.