Q. If the 2nd term of a GP is 8 and the 4th term is 32, what is the common ratio?
Solution
Let the first term be a and the common ratio be r. Then, 2nd term = ar = 8 and 4th term = ar^3 = 32. Dividing gives r^2 = 4, so r = 2.
Correct Answer: A — 2
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Q. If the 3rd term of a GP is 27 and the common ratio is 3, what is the first term?
Solution
Let the first term be a. Then, the 3rd term is ar^2 = 27. Thus, a * 3^2 = 27, giving a = 3.
Correct Answer: B — 9
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Q. If the first term of a GP is 7 and the common ratio is 3, what is the 6th term?
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A.
567
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B.
729
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C.
243
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D.
81
Solution
The 6th term is given by 7 * 3^(6-1) = 7 * 243 = 1701.
Correct Answer: B — 729
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Q. If the sum of the first three terms of a geometric progression is 14 and the common ratio is 2, what is the first term?
Solution
Let the first term be a. The sum of the first three terms is a + 2a + 4a = 7a. Setting 7a = 14 gives a = 2.
Correct Answer: B — 3
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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?
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 x and the common ratio is r, what is the expression for the sum of the first n terms?
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A.
x(1 - r^n)/(1 - r)
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B.
x(1 + r^n)/(1 + r)
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C.
xr^n/(1 - r)
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D.
xr^n/(1 + r)
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 GP, if the first term is 5 and the common ratio is 1/2, what is the sum of the first four terms?
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A.
15
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B.
10
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C.
12.5
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D.
20
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. Which of the following is NOT a property of geometric progressions?
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A.
The product of the terms is equal to the square of the geometric mean.
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B.
The sum of the terms can be negative.
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C.
The common ratio can be zero.
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D.
The terms can be fractions.
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?
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A.
1, 2, 4, 8
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B.
1, 3, 6, 10
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C.
2, 4, 8, 16
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D.
1, 1, 1, 1
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 geometric progressions is true?
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A.
The ratio of consecutive terms is constant.
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B.
The sum of terms is always positive.
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C.
The first term must be greater than the second.
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D.
The common ratio can only be an integer.
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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