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, you can use the formula:
an = a1 * r^(n-1)
Given a1 = 5 and r = 3, we can 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 find the equation for the nth term of a geometric sequence, you can use the formula:
an = a1 * r^(n-1)
Looking at the given geometric sequence -12, 4, -4/3, we can determine that a1 = -12 and r = 4/(-12) = -1/3. Substitute these values into the formula:
an = -12 * (-1/3)^(n-1)
Therefore, the correct answer is d) an = -12(-1/3)^(n-1).
3) To find the geometric means between two terms in a geometric sequence, you can use the formula:
m = sqrt(an * an-2)
Given the terms 5 and 1215, we need to find four geometric means in between. We can start by finding the first geometric mean:
m1 = sqrt(5 * 1215) = sqrt(6075) ≈ 77.849
Now we can continue finding the next geometric means using the formula:
m2 = sqrt(1215 * m1)
m3 = sqrt(m1 * m2)
m4 = sqrt(m2 * m3)
m5 = sqrt(m3 * m4)
Therefore, the correct answer is d) ±247, 489, ±731, 973.
4) To find the sum of a geometric series, you can use the formula:
S = a1 * (1 - r^n) / (1 - r)
Given the first term a1 = 128, the common ratio r = -1/2, and the number of terms n = 8, we can substitute these values into the formula:
S = 128 * (1 - (-1/2)^8) / (1 - (-1/2))
S = 128 * (1 - 1/256) / (1 + 1/2)
S = 128 * (1 - 1/256) / (3/2)
S = 128 * (255/256) / (3/2)
S = 128 * 255 / 768
S = 128 * 5/16
S = 640/16
S = 40
Therefore, the correct answer is a) 85.
5) The notation Σ (sigma) represents the sum of a series. The expression 5(-4)^(n-1) represents the terms in the series. The sigma notation can be written as:
Σ n=1 to 6 (5(-4)^(n-1))
To evaluate this sum, you can plug in the values of n from 1 to 6 and then add up the values. Let's calculate each term:
n = 1: 5(-4)^(1-1) = 5 * (-4)^0 = 5 * 1 = 5
n = 2: 5(-4)^(2-1) = 5 * (-4)^1 = 5 * (-4) = -20
n = 3: 5(-4)^(3-1) = 5 * (-4)^2 = 5 * 16 = 80
n = 4: 5(-4)^(4-1) = 5 * (-4)^3 = 5 * (-64) = -320
n = 5: 5(-4)^(5-1) = 5 * (-4)^4 = 5 * 256 = 1280
n = 6: 5(-4)^(6-1) = 5 * (-4)^5 = 5 * (-1024) = -5120
Now we can add up these values:
5 + (-20) + 80 + (-320) + 1280 + (-5120) = -4095
Therefore, the correct answer is b) -4095.