1A sequence of numbers is called an Arithmetic Progression (AP) if the difference between any two consecutive terms is:
A.Always zero
B.Always one
C.Constant
D.Variable
Correct Answer: Constant
Explanation:In an Arithmetic Progression, the difference between a term and its preceding term is always constant. This constant is called the common difference ().
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2Which of the following represents the -th term () of an Arithmetic Progression with first term and common difference ?
A.
B.
C.
D.
Correct Answer:
Explanation:The general term of an AP is given by the formula .
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3In a Geometric Progression (GP), the ratio of any term to its preceding term is called the:
A.Common difference
B.Common ratio
C.Common multiple
D.Common divisor
Correct Answer: Common ratio
Explanation:In a GP, consecutive terms have a constant ratio known as the common ratio, denoted by .
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4Which of the following is the formula for the -th term of a Geometric Progression?
A.
B.
C.
D.
Correct Answer:
Explanation:The general term of a GP with first term and common ratio is .
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5Three non-zero numbers are in Harmonic Progression (HP) if:
A.
B.
C. are in AP
D. are in GP
Correct Answer: are in AP
Explanation:By definition, a sequence is a Harmonic Progression if the reciprocals of its terms form an Arithmetic Progression.
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6Find the 10th term of the AP:
A.45
B.47
C.50
D.52
Correct Answer: 47
Explanation:Here , , and . Using : .
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7What is the sum of the first natural numbers?
A.
B.
C.
D.
Correct Answer:
Explanation:The sum of the first natural numbers is an AP with . The sum is .
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8If are in Arithmetic Progression, which relation holds true?
A.
B.
C.
D.
Correct Answer:
Explanation:Since the common difference is constant, . Rearranging gives .
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9Find the common ratio of the GP:
A.2
B.3
C.0.5
D.4
Correct Answer: 2
Explanation:The common ratio .
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10The sum of an infinite Geometric Progression exists only if the common ratio satisfies:
A.
B.
C.
D.
Correct Answer:
Explanation:An infinite geometric series converges to a finite sum only if the absolute value of the common ratio is less than 1 ().
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11Calculate the sum of the infinite GP:
A.1.5
B.2
C.2.5
D.Infinity
Correct Answer: 2
Explanation:Here and . Since , .
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12If , , and are the Arithmetic, Geometric, and Harmonic means of two distinct positive numbers, which inequality is correct?
A.
B.
C.
D.
Correct Answer:
Explanation:For any two positive real numbers, the Arithmetic Mean is greater than or equal to the Geometric Mean, which is greater than or equal to the Harmonic Mean.
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13What is the Geometric Mean (GM) of the numbers 4 and 16?
A.8
B.10
C.12
D.6
Correct Answer: 8
Explanation:.
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14What is the Arithmetic Mean (AM) of the numbers 4 and 16?
A.8
B.10
C.12
D.20
Correct Answer: 10
Explanation:.
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15The formula for the Harmonic Mean (HM) of two numbers and is:
A.
B.
C.
D.
Correct Answer:
Explanation:If is the HM of and , then are in HP, meaning are in AP. Solving for gives .
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16Which relation connects the Arithmetic Mean (A), Geometric Mean (G), and Harmonic Mean (H) of two numbers?
A.
B.
C.
D.
Correct Answer:
Explanation:Since , , and , calculating gives , which is .
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17Identify the type of progression:
A.Arithmetic Progression
B.Geometric Progression
C.Harmonic Progression
D.None of the above
Correct Answer: Geometric Progression
Explanation:The ratio between consecutive terms is constant (, ). Thus, it is a GP.
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18Identify the type of progression:
A.Arithmetic Progression
B.Geometric Progression
C.Harmonic Progression
D.Fibonacci Sequence
Correct Answer: Harmonic Progression
Explanation:Take the reciprocals: $2, 5, 8, 11$. This forms an AP with a common difference of 3. Therefore, the original sequence is an HP.
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19Find the sum of the first 20 terms of the AP:
A.590
B.610
C.570
D.600
Correct Answer: 590
Explanation:Using . Here . .
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20If are in Arithmetic Progression, find the value of .
A.3
B.4
C.2
D.-3
Correct Answer: 4
Explanation:For AP, . So, . .
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21What is the 5th term of the GP: ?
A.54
B.108
C.162
D.486
Correct Answer: 162
Explanation:Here . .
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22The sum of the first terms of a GP where is given by:
A.
B.
C.
D.
Correct Answer:
Explanation:When the common ratio is greater than 1, the formula for the sum is .
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23If every term of an AP is multiplied by a non-zero constant , the resulting sequence is:
A.An Arithmetic Progression
B.A Geometric Progression
C.A Harmonic Progression
D.Neither AP nor GP
Correct Answer: An Arithmetic Progression
Explanation:If the original terms are , the new terms are . This is an AP with common difference .
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24If every term of a GP is raised to the power 2, the resulting sequence is:
A.An Arithmetic Progression
B.A Geometric Progression
C.A Harmonic Progression
D.Constant
Correct Answer: A Geometric Progression
Explanation:If the original sequence is , the squared sequence is . This is a GP with common ratio .
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25Which term of the AP is 81?
A.18th
B.19th
C.20th
D.21st
Correct Answer: 20th
Explanation:. .
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26What is the common difference of the AP whose -th term is given by ?
A.5
B.3
C.8
D.2
Correct Answer: 3
Explanation:Find first two terms: , . Common difference . Generally, for a linear -th term , is the common difference.
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27Find the Harmonic Mean of 3 and 6.
A.4
B.4.5
C.5
D.2
Correct Answer: 4
Explanation:.
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28Which of the following sequences can never contain the term 0?
A.Arithmetic Progression
B.Geometric Progression
C.Harmonic Progression
D.Both GP and HP
Correct Answer: Both GP and HP
Explanation:In a GP, if a term is 0, the ratio is undefined or the sequence becomes trivial zeros. By definition of non-zero and , no term is 0. In HP, terms are reciprocals of an AP; since is undefined, 0 cannot be a term in HP.
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29What is the 7th term of the HP: ?
A.
B.
C.
D.
Correct Answer:
Explanation:Corresponding AP: (). 7th term of AP . Thus, 7th term of HP is .
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30The sum of the first odd natural numbers is:
A.
B.
C.
D.
Correct Answer:
Explanation:Sequence: (). .
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31If are in GP, then are in:
A.AP
B.GP
C.HP
D.None
Correct Answer: AP
Explanation:If (GP condition), taking logs: . This is the condition for AP.
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32Find if are in Geometric Progression.
A.1
B.2
C.3
D.4
Correct Answer: 2
Explanation:For GP, . .
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33The recurring decimal can be expressed as a sum of an infinite GP. The common ratio is:
A.0.3
B.0.1
C.0.01
D.0.9
Correct Answer: 0.1
Explanation: This is a GP with and .
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34If the sum of three numbers in AP is 15, the middle term is:
A.3
B.5
C.7
D.2.5
Correct Answer: 5
Explanation:Let terms be . Sum .
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35Which of the following is NOT an Arithmetic Progression?
A.1, 2, 3, 4
B.2, 4, 6, 8
C.5, 5, 5, 5
D.1, 3, 9, 27
Correct Answer: 1, 3, 9, 27
Explanation:1, 3, 9, 27 is a Geometric Progression with . The others are APs (including the constant sequence where ).
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36Find the sum of the series: to terms. (General approach)
A.
B.
C.
D.
Correct Answer:
Explanation:This is a standard problem type involving GP manipulation. . This results in the sum of a GP () minus .
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37If are in Harmonic Progression, then is called the:
A.Arithmetic Mean of and
B.Geometric Mean of and
C.Harmonic Mean of and
D.Common ratio
Correct Answer: Harmonic Mean of and
Explanation:The middle term of three numbers in HP is the Harmonic Mean of the outer two.
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38What is the 8th term of the sequence defined by ?
A.-16
B.16
C.-14
D.14
Correct Answer: 16
Explanation:.
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39Insert two Arithmetic Means between 5 and 11.
A.7, 9
B.6, 8
C.6, 9
D.7, 10
Correct Answer: 7, 9
Explanation:We need to be in AP. Total 4 terms. . . The terms are and .
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40If the -th term of a GP is , what is the first term ?
A.2
B.3
C.6
D.5
Correct Answer: 6
Explanation:.
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41In an AP, if , find the first term .
A.3
B.2
C.5
D.6
Correct Answer: 5
Explanation: represents the sum of the first 1 term, which is just the first term or . .
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42The product of three numbers in a Geometric Progression is 216. Find the middle term.
A.4
B.6
C.8
D.36
Correct Answer: 6
Explanation:Let the terms be . Product . Thus .
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43If and for two numbers, find their .
A.15
B.16
C.18
D.22.5
Correct Answer: 16
Explanation:Use . .
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44Which of the following is true for the sequence ?
A.It is an AP only
B.It is a GP only
C.It is both an AP and a GP
D.It is neither
Correct Answer: It is both an AP and a GP
Explanation:It is an AP with and a GP with .
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45Find the value of .
A.
B.
C.
D.
Correct Answer:
Explanation:This is an infinite GP with and . .
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46If denotes the sum of terms of an AP with common difference , then equals:
A.
B.
C.
D.$0$
Correct Answer:
Explanation:. So the expression becomes . By definition of AP, .
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47If positive numbers are in HP, then:
A.
B.
C. and
D.Cannot be determined without values
Correct Answer: Cannot be determined without values
Explanation:While HPs are monotonic (either increasing or decreasing) if terms are positive, we don't know if it's increasing or decreasing without specific values. However, usually lies strictly between and .
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48The sum of the first even natural numbers () is:
A.
B.
C.
D.
Correct Answer:
Explanation:.
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49Which term of the AP is -81?
A.34
B.35
C.36
D.33
Correct Answer: 35
Explanation:. .
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50If the 3rd term of a GP is 4, find the product of its first 5 terms.
A.256
B.1024
C.128
D.512
Correct Answer: 1024
Explanation:. Product of first 5 terms . Since , .
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