Q. In a geometric progression, if the 3rd term is 27 and the common ratio is 3, what is the first term?
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Solution
Let the first term be a. The 3rd term is a * r^2 = a * 3^2 = 9a. Setting 9a = 27 gives a = 3.
Correct Answer:
B
— 9
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Q. In a geometric progression, if the first term is 3 and the common ratio is 2, what is the 5th term?
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Solution
The nth term of a GP is given by a * r^(n-1). Here, a = 3, r = 2, and n = 5. Thus, the 5th term = 3 * 2^(5-1) = 3 * 16 = 48.
Correct Answer:
A
— 48
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Q. In a geometric progression, if the first term is 4 and the common ratio is 1/2, what is the 6th term?
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Solution
The 6th term is given by a * r^(n-1) = 4 * (1/2)^(6-1) = 4 * (1/32) = 4/32 = 0.125.
Correct Answer:
A
— 0.25
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Q. In a geometric progression, if the first term is 5 and the common ratio is 0.5, what is the sum of the first 4 terms?
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Solution
The sum of the first n terms of a GP is given by S_n = a(1 - r^n) / (1 - r). Here, S_4 = 5(1 - 0.5^4) / (1 - 0.5) = 5(1 - 0.0625) / 0.5 = 5 * 0.9375 / 0.5 = 9.375.
Correct Answer:
B
— 10
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Q. In a geometric progression, if the first term is 5 and the last term is 80 with 4 terms in total, what is the common ratio?
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Solution
The last term can be expressed as a * r^(n-1). Here, 80 = 5 * r^(4-1) = 5 * r^3. Thus, r^3 = 16, giving r = 2.
Correct Answer:
A
— 2
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Q. In a geometric progression, if the first term is 5 and the last term is 80, and there are 4 terms in total, what is the common ratio?
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Solution
Let the common ratio be r. The terms are 5, 5r, 5r^2, 5r^3. Setting 5r^3 = 80 gives r^3 = 16, thus r = 2.
Correct Answer:
A
— 2
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Q. In a geometric progression, if the first term is x and the common ratio is r, what is the expression for the sum of the first n terms?
A.
x(1 - r^n)/(1 - r)
B.
x(1 + r^n)/(1 + r)
C.
xr^n/(1 - r)
D.
xr^n/(1 + r)
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Solution
The sum of the first n terms of a GP is given by S_n = a(1 - r^n)/(1 - r) for r ≠ 1.
Correct Answer:
A
— x(1 - r^n)/(1 - r)
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Q. In a geometric progression, if the first term is x and the common ratio is y, what is the expression for the 3rd term?
A.
xy^2
B.
x/y^2
C.
x^2y
D.
x^2/y
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Solution
The 3rd term of a GP is given by a * r^(n-1). Here, it is x * y^(3-1) = xy^2.
Correct Answer:
A
— xy^2
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Q. In a GP, if the 3rd term is 27 and the 5th term is 243, what is the first term?
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Solution
Let the first term be a and the common ratio be r. Then, 3rd term = ar^2 = 27 and 5th term = ar^4 = 243. Dividing gives r^2 = 9, so r = 3. Substituting back gives a = 3.
Correct Answer:
B
— 9
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Q. In a GP, if the first term is 10 and the common ratio is 0.5, what is the 6th term?
A.
0.625
B.
1.25
C.
2.5
D.
5
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Solution
The 6th term is given by 10 * (0.5)^(6-1) = 10 * (0.5)^5 = 10 * 0.03125 = 0.3125.
Correct Answer:
A
— 0.625
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Q. In a GP, if the first term is 2 and the common ratio is -2, what is the 4th term?
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Solution
The 4th term is given by 2 * (-2)^(4-1) = 2 * (-8) = -16.
Correct Answer:
B
— -8
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Q. In a GP, if the first term is 4 and the common ratio is 1/2, what is the 6th term?
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Solution
The 6th term is given by a * r^(n-1) = 4 * (1/2)^(6-1) = 4 * (1/32) = 0.125, which is 0.25.
Correct Answer:
A
— 0.25
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Q. In a GP, if the first term is 5 and the common ratio is 1/2, what is the sum of the first four terms?
A.
15
B.
10
C.
12.5
D.
20
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Solution
The first four terms are 5, 2.5, 1.25, and 0.625. Their sum is 5 + 2.5 + 1.25 + 0.625 = 9.375.
Correct Answer:
C
— 12.5
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Q. In a GP, if the first term is 5 and the last term is 80, and there are 4 terms in total, what is the common ratio?
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Solution
Let the common ratio be r. The terms are 5, 5r, 5r^2, 5r^3. Setting 5r^3 = 80 gives r^3 = 16, thus r = 2.
Correct Answer:
A
— 2
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Q. In a GP, if the first term is 7 and the common ratio is 1/2, what is the 6th term?
A.
0.4375
B.
0.5
C.
1
D.
1.75
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Solution
The 6th term is given by 7 * (1/2)^(6-1) = 7 * (1/32) = 0.4375.
Correct Answer:
A
— 0.4375
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Q. In a GP, if the first term is x and the common ratio is y, what is the expression for the 6th term?
A.
xy^5
B.
xy^6
C.
x^6y
D.
x^5y
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Solution
The nth term of a GP is given by a * r^(n-1). Thus, the 6th term = x * y^(6-1) = xy^5.
Correct Answer:
A
— xy^5
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Q. What is the relationship between the first term and the common ratio if the sum of an infinite GP converges?
A.
The first term must be zero.
B.
The common ratio must be less than one in absolute value.
C.
The first term must be greater than the common ratio.
D.
The common ratio must be greater than one.
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Solution
For the sum of an infinite GP to converge, the common ratio must be less than one in absolute value.
Correct Answer:
B
— The common ratio must be less than one in absolute value.
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Q. What is the sum of the first 5 terms of a GP where the first term is 2 and the common ratio is 3?
A.
242
B.
364
C.
486
D.
728
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Solution
The sum of the first n terms of a GP is given by S_n = a(1 - r^n) / (1 - r). Here, S_5 = 2(1 - 3^5) / (1 - 3) = 2(1 - 243) / (-2) = 242.
Correct Answer:
A
— 242
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Q. Which of the following is NOT a characteristic of a geometric progression?
A.
The ratio of any two consecutive terms is constant.
B.
The product of the first and last terms equals the square of the middle term.
C.
The sum of the terms can be negative.
D.
The terms can be non-numeric.
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Solution
In a geometric progression, the terms are numeric and follow a specific ratio; thus, non-numeric terms do not fit the definition.
Correct Answer:
D
— The terms can be non-numeric.
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Q. Which of the following is NOT a property of a geometric progression?
A.
The product of the first and last terms equals the square of the geometric mean.
B.
The sum of the terms can be negative.
C.
The ratio of the last term to the first term is equal to the common ratio raised to the power of (n-1).
D.
The terms can be non-numeric.
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Solution
In a geometric progression, the terms are numeric and follow a specific ratio; thus, non-numeric terms do not form a valid GP.
Correct Answer:
D
— The terms can be non-numeric.
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Q. Which of the following is NOT a property of geometric progressions?
A.
The product of the terms is equal to the square of the geometric mean.
B.
The sum of the terms can be negative.
C.
The common ratio can be zero.
D.
The terms can be fractions.
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Solution
In a geometric progression, the common ratio cannot be zero, as it would invalidate the progression.
Correct Answer:
C
— The common ratio can be zero.
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Q. Which of the following sequences is a geometric progression?
A.
1, 2, 4, 8
B.
1, 3, 6, 10
C.
2, 4, 8, 16
D.
1, 1, 1, 1
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Solution
The sequence 2, 4, 8, 16 has a constant ratio of 2, making it a geometric progression.
Correct Answer:
C
— 2, 4, 8, 16
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Q. Which of the following statements about a geometric progression is true?
A.
The ratio of consecutive terms is constant.
B.
The difference between consecutive terms is constant.
C.
The sum of the terms is always positive.
D.
The first term is always the largest.
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Solution
In a geometric progression, the ratio of consecutive terms is indeed constant, which defines the progression.
Correct Answer:
A
— The ratio of consecutive terms is constant.
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Q. Which of the following statements about geometric progressions is true?
A.
The ratio of consecutive terms is constant.
B.
The sum of terms is always positive.
C.
The first term must be greater than the second.
D.
The common ratio can only be an integer.
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Solution
In a geometric progression, the ratio of consecutive terms is indeed constant, which defines the progression.
Correct Answer:
A
— The ratio of consecutive terms is constant.
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