1)Find the sixth term of the geometric sequence for which a1=5 and r=3
a.1215
b.3645
c.9375
d.23
answer=a
2)Write an equation for the nth term of the geometric sequence -12,4,-4/3,...
a.an=-12(1/3)n-1
b.an=12(-1/3)n-1
c.an=-12(-1/3)-n+1
d.an=-12(-1/3)n-1
answer=d
3)Find four geometric means between 5 and 1215
a.+-15,45,+-135,405
b.15,45,135,405
c.247,489,731,973
d.+-247,489,+-731,973
answer=d
4)Find the sum of the geometric series 128-64+32-_____to 8 terms
a.85
b.255
c.86
d.85/2
answer=a
5)Find 6 on top of sigma notation n=1 on the bottom 5(-4)n-1 on the right side.
a.6825
b.-4095
c.-1023
d.-5120
answer=b
1) To find the sixth term of a geometric sequence, we can use the formula:
an = a1 * r^(n-1)
Where:
an represents the nth term of the sequence,
a1 is the first term of the sequence,
r is the common ratio between the terms,
n is the position of the term we want to find.
Given that a1 = 5 and r = 3, we can substitute these values into the formula to find the sixth term:
a6 = 5 * 3^(6-1)
= 5 * 3^5
= 5 * 243
= 1215
Therefore, the sixth term of the geometric sequence is 1215. So, the correct answer is a) 1215.
2) To find the equation for the nth term of a geometric sequence, we can use the formula:
an = a1 * r^(n-1)
Where:
an represents the nth term,
a1 is the first term,
r is the common ratio,
n is the position of the term.
Given the geometric sequence -12, 4, -4/3, ..., we can see that the common ratio (r) is (-4/3) divided by (4), which is -1/3.
Substituting these values into the formula, we get:
an = -12 * (-1/3)^(n-1)
Therefore, the correct answer is d) -12(-1/3)^(n-1).
3) To find four geometric means between two numbers, we can use the formula for the nth term of a geometric sequence:
an = a1 * r^(n-1)
Where:
an represents the nth term,
a1 is the first term,
r is the common ratio,
n is the position of the term.
Given that the two numbers are 5 and 1215, we need to find r, the common ratio. Since we want four geometric means, the number of terms including the two given numbers is 6.
Using the formula, we have the following equation:
1215 = 5 * r^(6-1)
243 = r^5
r = 3
Now we can find the terms by substituting the values into the formula:
a2 = 5 * 3^(2-1) = 5 * 3 = 15
a3 = 5 * 3^(3-1) = 5 * 9 = 45
a4 = 5 * 3^(4-1) = 5 * 27 = 135
a5 = 5 * 3^(5-1) = 5 * 81 = 405
Therefore, the four geometric means between 5 and 1215 are: +-247, 489, +-731, 973. So, the correct answer is d) +-247, 489, +-731, 973.
4) To find the sum of a finite geometric series, we can use the formula:
Sn = a1 * (1 - r^n) / (1 - r)
Where:
Sn represents the sum of the first n terms,
a1 is the first term,
r is the common ratio,
n is the number of terms.
Given that the series is 128, -64, 32, ... and we need to find the sum of the first 8 terms, we can substitute the values into the formula:
Sn = 128 * (1 - (-0.5)^8) / (1 - (-0.5))
= 128 * (1 - 0.00390625) / (1 + 0.5)
= 128 * 0.99609375 / 1.5
= 85.333...
Rounding to the nearest whole number, the sum of the series is 85. So, the correct answer is a) 85.
5) The expression 6 on top of sigma notation, with n=1 at the bottom and 5(-4)^(n-1) on the right side, represents the sum of a geometric series.
Using the formula for the summation of a geometric series, we have:
S = a * (1 - r^n) / (1 - r)
Where:
S represents the sum of the series,
a is the first term of the series,
r is the common ratio,
n is the number of terms.
Given that a = 5, r = -4, and the number of terms is 6, we can substitute these values into the formula:
S = 5 * (1 - (-4)^6) / (1 - (-4))
= 5 * (1 - 4096) / (1 + 4)
= 5 * (-4095) / 5
= -4095
Therefore, the sum of the series is -4095. So, the correct answer is b) -4095.