use the first six terms to predict the limit of the sequence

a subtext n = n+6 / n^3 +6
(assume n begins with 1)

a. 1/6
b. 6
c. the sequence diverges
d. 0
e. 1

so, did you list the first 6 terms?

using inductive reasoning what is the prediction of the last two number in this set 1,-7,13,-19

I got the numbers 14,-26
is that right

To predict the limit of a sequence, we need to find the behavior of the terms as n approaches infinity.

Given the sequence an = (n+6) / (n^3 + 6), we can find the first six terms by substituting the values of n = 1, 2, 3, 4, 5, and 6.

a1 = (1+6) / (1^3 + 6) = 7 / 7 = 1
a2 = (2+6) / (2^3 + 6) = 8 / 14
a3 = (3+6) / (3^3 + 6) = 9 / 33
a4 = (4+6) / (4^3 + 6) = 10 / 70
a5 = (5+6) / (5^3 + 6) = 11 / 131
a6 = (6+6) / (6^3 + 6) = 12 / 216

Now, let's examine the behavior of these terms:

a1 = 1/1 = 1
a2 ≈ 0.571
a3 ≈ 0.273
a4 ≈ 0.143
a5 ≈ 0.084
a6 ≈ 0.056

Based on the values of the terms, we can see that as n increases, the terms are approaching zero. This suggests that the limit of the sequence as n approaches infinity is 0.

Therefore, the answer is option d. 0.