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 that a1 = 5 and r = 3, we can substitute these values into the formula:
a6 = 5 * 3^(6-1)
= 5 * 3^5
= 5 * 243
= 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 that the sequence is -12, 4, -4/3, ..., we can see that a1 = -12 and r = 4/(-12) = -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 can use the formula:
gm = sqrt(a * b)
Given that a = 5 and b = 1215, we can find the first geometric mean:
gm1 = sqrt(5 * 1215)
= sqrt(6075)
= 77.98 (approximately)
Then, we can find the second geometric mean by using gm1 as a and b:
gm2 = sqrt(77.98 * 1215)
= sqrt(94622.7)
= 307.93 (approximately)
Similarly, we can find the third and fourth geometric means:
gm3 = sqrt(307.93 * 1215)
= sqrt(374040.95)
= 611.99 (approximately)
gm4 = sqrt(611.99 * 1215)
= sqrt(743536.85)
= 862.87 (approximately)
Therefore, the correct answer is (d) ±247, 489, ±731, 973.
4) To find the sum of a geometric series, we can use the formula:
Sn = a1 * (1 - r^n) / (1 - r)
Given that a1 = 128, r = -1/2, and n = 8, we can substitute these values into the formula:
S8 = 128 * (1 - (-1/2)^8) / (1 - (-1/2))
= 128 * (1 - 1/256) / (3/2)
= 128 * (255/256) / (3/2)
= 128 * (255/256) * (2/3)
= 85
Therefore, the correct answer is (a) 85.
5) The given expression can be written in sigma notation as follows:
Σ(5 * (-4)^(n-1)), from n = 1 to 6
Expanding the series, we get:
5 * (-4)^(1-1) + 5 * (-4)^(2-1) + 5 * (-4)^(3-1) + 5 * (-4)^(4-1) + 5 * (-4)^(5-1) + 5 * (-4)^(6-1)
Simplifying each term, we get:
5 * (-4)^0 + 5 * (-4)^1 + 5 * (-4)^2 + 5 * (-4)^3 + 5 * (-4)^4 + 5 * (-4)^5
= 5 * 1 + 5 * (-4) + 5 * 16 + 5 * (-64) + 5 * 256 + 5 * (-1024)
= 5 - 20 + 80 - 320 + 1280 - 5120
= -4095
Therefore, the correct answer is (b) -4095.
To find the sixth term of a geometric sequence with a1 = 5 and r = 3, we can use the formula for the nth term of a geometric sequence: an = a1 * r^(n-1).
Substituting the given values, we have a6 = 5 * 3^(6-1) = 5 * 3^5 = 5 * 243 = 1215. Thus, the answer is a.
To write an equation for the nth term of the geometric sequence -12, 4, -4/3, ..., we need to determine the common ratio (r). We can do this by dividing any term by its preceding term. In this case, r = (4) / (-12) = -1/3.
Using this value of r, the equation for the nth term is: an = (-12) * (-1/3)^(n-1). Simplifying this expression, we get an = -12 * (-1/3)^(n-1). Thus, the answer is d.
To find four geometric means between 5 and 1215, we need to determine the common ratio (r). In this case, r = √(1215/5) = √243 = ±3.
Using this value of r, the geometric means can be found by multiplying the previous term by the common ratio. Starting with 5, the sequence is 5, ±15, 45, ±135, 405, ±1215. Thus, the answer is d.
To find the sum of the geometric series 128 - 64 + 32 - ..., we can use the formula for the sum of a finite geometric series: S = a * (1 - r^n) / (1 - r), where S is the sum, a is the first term, r is the common ratio, and n is the number of terms.
In this case, a = 128, r = -1/2, and n = 8. Substituting these values, we have S = 128 * (1 - (-1/2)^8) / (1 - (-1/2)) = 128 * (1 - 1/256) / (3/2) = 128 * (255/256) / (3/2) = 128 * 255 / (256 * 3) = 85. Thus, the answer is a.
The expression 6 on top of sigma notation with n = 1 on the bottom and 5(-4)^(n-1) on the right side represents the sum of a geometric series.
Expanding the terms, we have 5(-4)^(1-1) + 5(-4)^(2-1) + 5(-4)^(3-1) + 5(-4)^(4-1) + 5(-4)^(5-1) + 5(-4)^(6-1). Simplifying this expression, we get 5(-4)^0 + 5(-4)^1 + 5(-4)^2 + 5(-4)^3 + 5(-4)^4 + 5(-4)^5 = 5 + (-20) + 80 + (-320) + 1280 + (-5120) = -4095. Thus, the answer is b.