1) To find the sixth term of a geometric sequence, we can 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 = 5 and r = 3, we substitute these values into the formula:
a6 = 5 * 3^(6-1)
a6 = 5 * 3^5
a6 = 5 * 243
a6 = 1215
Therefore, the correct answer is A) 1215.
2) To write an equation for the nth term of a geometric sequence, we can use the formula an = a1 * r^(n-1).
Given the geometric sequence -12, 4, -4/3, we know that the first term a1 = -12. In order to find the common ratio, we can divide any term by the previous term. In this case, (-4/3) / 4 = -1/3. Therefore, the common ratio r = -1/3.
Substituting these values into the formula, we get:
aN = -12 * (-1/3)^(n-1)
Therefore, the correct answer is D) aN = -12(-1/3)^(n-1).
3) To find four geometric means between 5 and 1215, we need to find the common ratio (r) of the geometric sequence. The formula to find the common ratio is r = (term2/term1)^(1/(n-1)), where term2 is the second term, term1 is the first term, and n is the total number of terms (including the first and last terms).
Given that term1 = 5, term2 = 1215, and n = 6 (including the first and last terms), we substitute these values into the formula:
r = (1215/5)^(1/(6-1))
r = 243^(1/5)
r = 3
Now, we can find the four geometric means using the formula an = a1 * r^(n-1).
First geometric mean:
a2 = 5 * 3^(2-1)
a2 = 15
Second geometric mean:
a3 = 15 * 3^(3-1)
a3 = 45
Third geometric mean:
a4 = 45 * 3^(4-1)
a4 = 135
Fourth geometric mean:
a5 = 135 * 3^(5-1)
a5 = 405
Therefore, the correct answer is B) 15, 45, 135, 405.
4) To find the sum of a geometric series with a given number of terms, we can use the formula S = a1 * (1 - r^n) / (1 - r), where S is the sum of the series, a1 is the first term, r is the common ratio, and n is the number of terms.
Given the geometric series 128, -64, 32, ..., we can see that the first term a1 = 128 and the common ratio r = -1/2. We are asked to find the sum of the series to 8 terms, so n = 8.
Substituting these values into the formula, we get:
S = 128 * (1 - (-1/2)^8) / (1 - (-1/2))
S = 128 * (1 - 1/256) / (1 + 1/2)
S = 128 * (255/256) / (3/2)
S = (128 * 255 * 2) / (256 * 3)
S = 85
Therefore, the correct answer is A) 85.
5) In the given expression ∑6n=15(-4)^(n-1), we want to find the value.
Using the formula for the sum of a geometric series, we have:
S = a * (1 - r^n) / (1 - r)
In this case, a = 5, r = -4, and n ranges from 1 to 6.
Substituting these values into the formula, we get:
S = 5 * (1 - (-4)^6) / (1 - (-4))
Now, we evaluate this expression:
S = 5 * (1 - 4096) / (1 + 4)
S = 5 * (-4095) / 5
S = -4095
Therefore, the correct answer is B) -4095.