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
since the terms are number an, then n is any integer from one onward. n>=1.
since a4 = a1*r^3,
1024r^3 = -16
r^3 = -16/1024 = -2^4/2^10 = -2^-6
So, r = -2^-2 = -1/4
a6 = a4*r^2 = -16 * 1/16 = -1
Hhb
1) To find the domain for n in the arithmetic sequence an = 4 - 3(n - 1), we need to determine the valid values for n.
In this sequence, the formula represents the nth term of the sequence. The value of n determines which term we are finding.
The formula starts with n - 1, which means that n should be at least 1 to have a valid term. If n is less than 1, the formula would involve negative indexing, which doesn't make sense in this case.
So the correct answer is: All integers where n ≥ 1.
2) To find the 6th term of a geometric sequence with given terms a1 = 1,024 and a4 = -16, we need to determine the common ratio (r) first.
A geometric sequence has the form: an = a1 * r^(n-1).
We can find the common ratio (r) using a4/a1.
a4 / a1 = -16 / 1,024 = -1/64.
So the common ratio (r) is -1/64.
Now we can use the formula to find the 6th term:
a6 = a1 * r^(6-1)
= 1,024 * (-1/64)^5
To simplify this, we can rewrite (-1/64)^5 as (-1)^5 / (64)^5.
a6 = 1,024 * (-1)^5 / (64)^5
= 1,024 * (-1) / (64^5)
= -1,024 / (1,048,576)
a6 = -1/1,024
So, the 6th term of the geometric sequence is: -0.25.
1) To find the domain for n in the given arithmetic sequence an = 4 - 3(n - 1), we need to determine the range of values that n can take. In an arithmetic sequence, the terms are generated by adding a constant difference (d) to the previous term. In this case, the constant difference is -3.
To determine the domain, we need to find the value of n that satisfies the given condition. In this case, we have the formula an = 4 - 3(n - 1).
At the first term (n = 1), an = 4 - 3(1 - 1) = 4 - 3(0) = 4. So, the sequence starts at n = 1.
To find the domain, we need to determine how far the sequence can go. It depends on the values that n can take. Since there are no limitations mentioned in the given sequence, we can assume that n can take any positive integer value.
Therefore, the domain for n in this arithmetic sequence is "All integers where n ≥ 1."
2) To find the 6th term of the geometric sequence, we need to determine the common ratio (r) of the sequence. In a geometric sequence, each term is generated by multiplying the previous term by a constant factor (r).
Given that a1 = 1,024 and a4 = -16, we can write the ratio between the terms as:
r = a4 / a1
r = -16 / 1,024
To simplify the ratio, we can divide both numerator and denominator by 16:
r = -1 / 64
Now that we have the common ratio (r = -1 / 64), we can find the 6th term using the general formula for geometric sequences:
an = a1 * r^(n-1)
a6 = a1 * r^(6-1)
a6 = 1,024 * (-1 / 64)^(5)
It is essential to note that raising a negative number to an odd exponent will result in a negative number.
a6 = 1,024 * (-1/64)^5
Simplifying further,
a6 = 1,024 * (-1)^5 / (64^5)
Since (-1)^5 = -1, and 64^5 = 1,073,741,824,
a6 = 1,024 * (-1) / 1,073,741,824
Finally, calculating the value,
a6 = -1,024 / 1,073,741,824
Therefore, the 6th term of the given geometric sequence is approximately -0.0000009537, which can be rounded to -1 (-1.0) for practical purposes.
So, the 6th term of the geometric sequence is -1.