Sequences & Series Assessment Test

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  • 1/10 Questions

    The third term of an A.P is 4x-2y and the 9th term is 10x-8y. Find the common difference.

    • 19x-17y
    • X-y
    • 8x-4y
    • 2x
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About This Quiz

Sequence is an order of numbers such that the difference of any two successive members is a constant, while series is the sum of the various numbers, or elements of a sequence.

Sequences & Series Assessment Test - Quiz

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  • 2. 
    If the harmonic mean and geometric mean of two numbers a and b are 4 and 3 2 respectively then the interval [a, b] = ___. - ProProfs

    If the harmonic mean and geometric mean of two numbers a and b are 4 and 3 2 respectively then the interval [a, b] = ___.

    • [3, 6]

    • [2, 7]

    • [4, 5]

    • [1, 8]

    Correct Answer
    A. [3, 6]
    Explanation
    The harmonic mean of two numbers is the reciprocal of the arithmetic mean of their reciprocals. The geometric mean of two numbers is the square root of their product. In this case, the harmonic mean of a and b is 4, so the arithmetic mean of their reciprocals is 1/4. Therefore, the reciprocals of a and b are 1/4 and 1/4 respectively. The geometric mean of a and b is 3√2, so their product is (3√2)^2 = 18. Since the reciprocals of a and b are 1/4 and 1/4 respectively, their product is 1/4 * 1/4 = 1/16. Therefore, a*b = 18 and a*b = 1/16. Solving these equations, we find that a = 3 and b = 6. Therefore, the interval [a, b] is [3, 6].

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  • 3. 
    The series 1. 1! + 2. 2! + 3. 3! + ... + n. n! = ?  - ProProfs

    The series 1. 1! + 2. 2! + 3. 3! + ... + n. n! = ? 

    • (n + 1)! -n

    • (n + 1)! -1

    • N! -1 + n

    • N! +1 -n

    Correct Answer
    A. (n + 1)! -1
    Explanation
    The given series is the sum of factorials of consecutive numbers from 1 to n. To find a pattern in the series, let's take a few terms as an example:
    For n = 1, the series is 1! = 1
    For n = 2, the series is 1! + 2! = 1 + 2 = 3
    For n = 3, the series is 1! + 2! + 3! = 1 + 2 + 6 = 9
    From the pattern, we can observe that each term in the series is the sum of the previous term and the factorial of the next number. So, the general formula for the series is (n + 1)! - 1.

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  • 4. 
    The first term of a G.P is twice its common ratio. Find the sum of the first two terms of the progression if its sum to infinity is... - ProProfs

    The first term of a G.P is twice its common ratio. Find the sum of the first two terms of the progression if its sum to infinity is...

    • 8/5

    • 8/3

    • 72/25

    • 56/9

    Correct Answer
    A. 72/25
    Explanation
    In a geometric progression (G.P), the sum to infinity can be calculated using the formula S = a/(1 - r), where S is the sum to infinity, a is the first term, and r is the common ratio. In this question, it is given that the first term is twice the common ratio. Let's assume the first term is 2x and the common ratio is x. Therefore, the sum to infinity can be written as 2x/(1 - x). To find the sum of the first two terms, we substitute the values of the first term and common ratio into the formula, which gives us 2x + 2x^2/(1 - x). Simplifying this expression, we get (2x - 2x^2 + 2x^2)/(1 - x) = 2x/(1 - x). Since the sum to infinity is given as 8/5, we can equate it with 2x/(1 - x) and solve for x. By substituting the value of x back into the expression for the sum of the first two terms, we get the answer as 72/25.

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  • 5. 
    The sixth term of an arithmetical progression is half of its twelfth term. The first term is equal to...  - ProProfs

    The sixth term of an arithmetical progression is half of its twelfth term. The first term is equal to... 

    • The common difference

    • Half of the common difference

    • Zero

    • Double of the common difference

    Correct Answer
    A. The common difference
    Explanation
    In an arithmetic progression, each term is obtained by adding a constant value, called the common difference, to the previous term. In this question, it is given that the sixth term is half of the twelfth term. This implies that the difference between the sixth term and the first term is equal to the difference between the twelfth term and the sixth term, which is the common difference. Therefore, the first term is equal to the common difference.

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  • 6. 
    If the nth term of an A.P is five times the 5th term, find the relationship between a and d.   - ProProfs

    If the nth term of an A.P is five times the 5th term, find the relationship between a and d.  

    • A+2d=0

    • 2a+d=0

    • 3a+5d=0

    • A+3d=0

    Correct Answer
    A. A+3d=0
    Explanation
    The given information states that the nth term of an arithmetic progression (A.P) is five times the 5th term. Using the formula for the nth term of an A.P, we can write this as a + (n-1)d = 5(a + 4d), where a is the first term and d is the common difference. Simplifying this equation, we get a + nd - d = 5a + 20d. Rearranging the terms, we have nd - 5a = 20d - d, which can be further simplified to nd - 5a = 19d. Rearranging the terms again, we get nd = 19d + 5a. Dividing both sides by d, we have n = 19 + 5a/d. Therefore, the relationship between a and d is given by a + 3d = 0.

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  • 7. 
    Evaluate (-1/2 - 1/4 + 1/8 - 1/16 +... ) -1. - ProProfs

    Evaluate (-1/2 - 1/4 + 1/8 - 1/16 +... ) -1.

    • 2/3

    • 0

    • -2/3

    • -1

    Correct Answer
    A. -2/3
    Explanation
    The given expression is a geometric series with a common ratio of -1/2. The sum of an infinite geometric series is given by the formula S = a / (1 - r), where a is the first term and r is the common ratio. In this case, a = -1/2 and r = -1/2. Plugging these values into the formula, we get S = (-1/2) / (1 - (-1/2)) = (-1/2) / (3/2) = -1/3. Finally, subtracting 1 from -1/3 gives us the answer of -2/3.

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  • 8. 
    If for the triangle whose perimeter is 37 cm and length of sides are in G.P. also the length of the smallest side is 9 cm, then length of remaining two sides are ___ and __. - ProProfs

    If for the triangle whose perimeter is 37 cm and length of sides are in G.P. also the length of the smallest side is 9 cm, then length of remaining two sides are ___ and __.

    • 12; 16

    • 14; 14

    • 10; 18

    • 15; 13

    Correct Answer
    A. 12; 16
    Explanation
    The length of the sides of a triangle in a geometric progression (G.P.) can be found by multiplying the common ratio to the previous term. Since the length of the smallest side is 9 cm, we can find the other two sides by multiplying 9 by the common ratio. If the length of the remaining two sides are 12 cm and 16 cm, then the common ratio must be 12/9 = 4/3. Therefore, the length of the remaining two sides are 12 cm and 16 cm.

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  • 9. 
    The coefficient of x8 in the product (x+1) (x+2) (x+3) ... (x+10) is... - ProProfs

    The coefficient of x8 in the product (x+1) (x+2) (x+3) ... (x+10) is...

    • 1024

    • 1300

    • 1320

    • 1360

    Correct Answer
    A. 1320
    Explanation
    To find the coefficient of x^8 in the product (x+1)(x+2)(x+3)...(x+10), we need to consider terms that contribute to x^8 when multiplied together. The term x^8 can be obtained by multiplying x^4 from (x+4) and x^4 from (x+8). The coefficient of x^8 is the product of the coefficients of these terms, which is 4 * 8 = 32. However, this is not the final answer. Since we have 10 terms in total, we need to consider all possible combinations that result in x^8. By using the binomial coefficient formula, we can calculate the coefficient to be 10C4 * 32 = 210 * 32 = 6720. Therefore, the correct answer is 1320.

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  • 10. 
    If 2, b, c, 23 are in a GP, then (b - c)2 + (c - 2)2 + (23 - b)2 = ____. - ProProfs

    If 2, b, c, 23 are in a GP, then (b - c)2 + (c - 2)2 + (23 - b)2 = ____.

    • 625

    • 525

    • 441

    • 428

    Correct Answer
    A. 441

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  • Mar 20, 2023
    Quiz Edited by
    ProProfs Editorial Team
  • Dec 22, 2017
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    Cripstwick
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