1) Given the arithmetic sequence an = 4 - 3(n - 1), what is the domain for n?
All integers where n ≥ 1
All integers where n > 1
All integers where n ≤ 4
All integers where n ≥ 4
2) What is the 6th term of the geometric sequence where a1 = 1,024 and a4 = -16?
1
-0.25
-1
0.25
To determine the domain for n in the given arithmetic sequence an = 4 - 3(n - 1), we need to find the values that n can take.
In an arithmetic sequence, the common difference between consecutive terms remains constant. In this case, the common difference is -3.
To find the domain for n, we can consider the formula for an arithmetic sequence given by:
an = a1 + (n - 1)d,
where a1 is the first term and d is the common difference.
Comparing this formula with the given sequence an = 4 - 3(n - 1), we can see that a1 = 4 and d = -3.
Now, let's consider the conditions given in the answer choices:
1) All integers where n ≥ 1:
The sequence is defined for all values of n starting from 1 and onwards.
2) All integers where n > 1:
This condition excludes n = 1.
3) All integers where n ≤ 4:
This condition limits the values of n up to 4, including 4 but excluding anything greater.
4) All integers where n ≥ 4:
This condition includes the values of n starting from 4 and onwards.
Based on the formula for an arithmetic sequence and the given sequence, we can conclude that the correct answer is:
All integers where n ≥ 1.
Now, let's move on to the second question.
To find the 6th term of a geometric sequence, we need to use the formula:
an = a1 * r^(n - 1),
where a1 is the first term, r is the common ratio, and n is the term number.
Given that a1 = 1,024 and a4 = -16, we can use this information to find the common ratio:
a4 = a1 * r^(4 - 1),
-16 = 1,024 * r^3.
Dividing both sides of the equation by 1,024, we have:
-16/1,024 = r^3,
-1/64 = r^3.
Taking the cube root of both sides, we get:
r = -1/4.
Now that we know the common ratio, we can find the 6th term:
a6 = a1 * r^(6 - 1),
a6 = 1,024 * (-1/4)^5.
Calculating this expression, we get:
a6 = 1,024 * (-1/1024),
a6 = -1.
Therefore, the 6th term of the geometric sequence is:
-1.