PRC-2 · Chapter 3

Mathematical Progression MCQs with Answers

45 multiple-choice questions on Mathematical Progression for PRC-2 Quantitative Analysis for Business. Try each one before revealing the answer and explanation.

1. Question 1

   What is the specific mathematical term for a succession of numbers formed and arranged according to a definite law or rule?

   • A) Coordinate
   • B) Series
   • C) Progression
   • D) Variable
   Show answer & explanation

   Answer: B) Series

   A series, or sequence, is formally defined in mathematics as the succession of terms formed and arranged according to some definite law or rule.

2. Question 2

   In an arithmetic progression, which mathematical formula is used to find the value of the 'nth' term?

   • A) a + (n - 1)d
   • B) a * r^(n-1)
   • C) n/2 * (2a + (n-1)d)
   • D) a / (1 - r)
   Show answer & explanation

   Answer: A) a + (n - 1)d

   The value of any specific term in an arithmetic progression can be found using the formula nth term = a + (n - 1)d, where 'a' is the first term and 'd' is the common difference.

3. Question 3

   Find the 8th term of an arithmetic progression where the first term is 10 and the common difference is 5.

   • A) 40
   • B) 45
   • C) 50
   • D) 55
   Show answer & explanation

   Answer: B) 45

   Using the arithmetic term formula a + (n-1)d, we substitute the values: 10 + (8-1)*5 = 10 + 35 = 45.

4. Question 4

   What is the total sum of the first 8 terms of an arithmetic progression if the first term is 10 and the common difference is 5?

   • A) 180
   • B) 200
   • C) 220
   • D) 240
   Show answer & explanation

   Answer: C) 220

   The sum formula is S = (n/2) * [2a + (n-1)d]. Substituting the knowns: (8/2) * [2(10) + (8-1)5] = 4 * [20 + 35] = 4 * 55 = 220.

5. Question 5

   In a geometric progression, how is the relationship between consecutive terms defined?

   • A) By adding a constant number to the previous term.
   • B) By squaring the value of the previous term.
   • C) By multiplying the previous term by a constant ratio.
   • D) By subtracting a constant difference.
   Show answer & explanation

   Answer: C) By multiplying the previous term by a constant ratio.

   A geometric progression is mathematically formed when each subsequent term in the sequence is linked by multiplying the previous term by a constant, known as the common ratio.

6. Question 6

   What is the 6th term of the following geometric progression: 1000, 800, 640...?

   • A) 512
   • B) 409.6
   • C) 327.68
   • D) 262.14
   Show answer & explanation

   Answer: C) 327.68

   The common ratio 'r' is found by 800/1000 = 0.8. The 6th term is calculated as a*r^(n-1) = 1000 * (0.8)^5 = 1000 * 0.32768 = 327.68.

7. Question 7

   Under what condition can the sum of an infinite geometric series be calculated using the convergence formula S = a / (1 - r)?

   • A) When r is positive and greater than 1.
   • B) When r is a negative number less than -1.
   • C) When the common ratio r is between -1 and 1.
   • D) Only when the common ratio r is exactly 0.
   Show answer & explanation

   Answer: C) When the common ratio r is between -1 and 1.

   As the number of terms approaches infinity, the sequence's sum can only converge to a finite value using S = a / (1 - r) if the common ratio 'r' is a fraction between -1 and 1.

8. Question 8

   Find the sum of the first 12 terms of a geometric progression where the first term is 5 and the common ratio is -2.

   • A) -6,825
   • B) 6,825
   • C) -13,650
   • D) 13,650
   Show answer & explanation

   Answer: A) -6,825

   Using the geometric sum formula S = a(1 - r^n) / (1 - r): 5(1 - (-2)^12) / (1 - (-2)) = 5(1 - 4096) / 3 = 5(-4095) / 3 = -20475 / 3 = -6,825.

9. Question 9

   If every single term in an arithmetic progression is multiplied by a constant 'k', what happens to the new common difference?

   • A) It remains the same as the original difference.
   • B) It increases cumulatively by k.
   • C) It is multiplied by the constant k.
   • D) It is divided by the constant k.
   Show answer & explanation

   Answer: C) It is multiplied by the constant k.

   According to the structural properties of arithmetic progressions, if each term is multiplied by a constant k, the new common difference becomes the original common difference multiplied by k.

10. Question 10

    A manufacturer sells 1000 units in January, but sales decrease by half each month thereafter. Which formula calculates the total sales for the first 5 months?

    • A) S = 1000(1 - 0.5^5) / (1 - 0.5)
    • B) S = 1000(1 - 2^5) / (1 - 2)
    • C) S = (5/2)[2(1000) + 4(0.5)]
    • D) S = 1000 * 0.5^4
    Show answer & explanation

    Answer: A) S = 1000(1 - 0.5^5) / (1 - 0.5)

    Because the sales halve each month, this represents a geometric progression with a = 1000 and r = 0.5. The appropriate sum formula is S = a(1 - r^n) / (1 - r).

11. Question 11

    If the sum of three consecutive terms in an Arithmetic Progression totals 54, what is the exact value of the middle term?

    • A) 15
    • B) 18
    • C) 21
    • D) 24
    Show answer & explanation

    Answer: B) 18

    Let the three terms be represented as (a - d), a, and (a + d). Their total sum is (a - d) + a + (a + d) = 3a = 54. Therefore, the middle term 'a' must be 54 / 3 = 18.

12. Question 12

    The first and last terms of an arithmetic progression with exactly three terms are X and -X respectively. What is the common difference?

    • A) X
    • B) -X
    • C) 0
    • D) 2X
    Show answer & explanation

    Answer: B) -X

    If the terms are X, a2, and -X. The common difference 'd' can be stated as d = a2 - X and d = -X - a2. Adding them gives 2d = -2X, meaning d = -X.

13. Question 13

    The second term of a given geometric progression is exactly half of the first term. If the first term is denoted by 'a', what is the common ratio 'r'?

    • A) 1/2
    • B) 2
    • C) 1/4
    • D) 4
    Show answer & explanation

    Answer: A) 1/2

    In a geometric progression, the second term is calculated as a*r. Since the problem states a*r = a/2, solving for the common ratio 'r' yields 1/2.

14. Question 14

    Which of the following is the standard shortcut formula used for finding the sum of the first 'n' natural numbers (1 + 2 + 3 + ... + n)?

    • A) n(n + 1)/2
    • B) n(n - 1)/2
    • C) (n + 2)/2
    • D) n^2/2
    Show answer & explanation

    Answer: A) n(n + 1)/2

    A sequence of natural numbers is an arithmetic series where a=1 and d=1. Using the formula S = n/2(2a + (n-1)d) results in S = n/2(2 + n - 1) = n(n + 1)/2.

15. Question 15

    In an arithmetic progression, the first term is 10 and the common difference is 2. Which equation accurately models finding the number of terms 'n' needed to produce a sum of 252?

    • A) 252 = n(20 + (n-1)2) / 2
    • B) 252 = 10 * 2^(n-1)
    • C) 252 = 10 + (n-1)2
    • D) 252 = n(10 + 2n)
    Show answer & explanation

    Answer: A) 252 = n(20 + (n-1)2) / 2

    Using the arithmetic sum formula Sn = (n/2)[2a + (n-1)d]. Substituting the specific values given (Sn=252, a=10, d=2) results in the equation 252 = (n/2)[20 + (n-1)2].

16. Question 16

    A tech company develops a new product projected to earn Rs. 150 million in revenue during its first year. If revenue strictly declines by 25% each subsequent year, what is the total projected revenue over the infinite life of the product?

    • A) Rs. 400 million
    • B) Rs. 500 million
    • C) Rs. 600 million
    • D) Rs. 750 million
    Show answer & explanation

    Answer: C) Rs. 600 million

    This forms an infinite geometric progression where a = 150. A 25% decline means the common ratio 'r' is 1 - 0.25 = 0.75. Using the sum to infinity formula S = a / (1 - r) yields 150 / 0.25 = 600 million.

17. Question 17

    In a geometric progression, the third term is exactly 80 and the common ratio is 4. What is the value of the first term?

    • A) 4
    • B) 5
    • C) 10
    • D) 20
    Show answer & explanation

    Answer: B) 5

    Using the nth term formula for a GP, a*r^(n-1). For the 3rd term, a*(4)^2 = 80. This simplifies to 16a = 80, meaning a = 5.

18. Question 18

    In an arithmetic progression, the first term is 12 and the common difference is 4. How many specific terms are needed to produce a total sum of exactly 320?

    • A) 8
    • B) 10
    • C) 12
    • D) 14
    Show answer & explanation

    Answer: B) 10

    Using the formula Sn = (n/2)[2a + (n-1)d]: 320 = (n/2)[24 + (n-1)4]. This expands to 640 = n(20 + 4n) -> 4n^2 + 20n - 640 = 0 -> n^2 + 5n - 160 = 0. Factoring yields (n-10)(n+16)=0. Thus, n=10.

19. Question 19

    A person's savings plan is structured as a geometric progression. They save Rs. 200 in month 1, Rs. 100 in month 2, Rs. 50 in month 3, and so on. What is the total amount saved at the end of exactly 5 months?

    • A) Rs. 387.5
    • B) Rs. 390.0
    • C) Rs. 395.5
    • D) Rs. 400.0
    Show answer & explanation

    Answer: A) Rs. 387.5

    This is a finite GP sum where a = 200, r = 0.5, and n = 5. Using S = a(1 - r^n) / (1 - r): S = 200(1 - 0.5^5) / 0.5 = 400(1 - 0.03125) = 400(0.96875) = Rs. 387.5.

20. Question 20

    The first and last terms of an arithmetic progression are 20 and 100, respectively. If the total sum of all terms is 300, what is the exact common difference?

    • A) 15
    • B) 20
    • C) 25
    • D) 30
    Show answer & explanation

    Answer: B) 20

    First, find n using S = (n/2)(a + l): 300 = (n/2)(20 + 100) -> 300 = 60n -> n = 5. Then, use the last term formula l = a + (n-1)d: 100 = 20 + 4d -> 80 = 4d -> d = 20.

21. Question 21

    A retailer sells 1,024 units of an obsolete product in January. Sales volume is strictly halved each month thereafter. The product is discontinued entirely after the month where only 1 unit is sold. How many total units are sold over this period?

    • A) 2,000
    • B) 2,046
    • C) 2,048
    • D) 4,096
    Show answer & explanation

    Answer: B) 2,046

    This is a GP where a=1024 and r=0.5. The sequence ends at 1, which represents 11 total terms (1024 * 0.5^10 = 1). The sum S = 1024(1 - 0.5^11) / 0.5 = 2048(1 - 1/2048) = 2048 - 1 = 2046 units.

22. Question 22

    A business deposits Rs. 5,000 into a fund in year 1. They decide to increase the annual deposit by exactly 20% each subsequent year. What is the precise deposit amount required in the 5th year?

    • A) Rs. 9,000
    • B) Rs. 10,368
    • C) Rs. 12,441.6
    • D) Rs. 14,929.9
    Show answer & explanation

    Answer: B) Rs. 10,368

    A constant percentage increase forms a geometric progression. Here a = 5000, r = 1.20, and n = 5. The 5th term is a*r^4 = 5000 * (1.2)^4 = 5000 * 2.0736 = Rs. 10,368.

23. Question 23

    In a long-term contract structure representing an arithmetic progression, the 8th payment is Rs. 18,000 and the 12th payment is Rs. 26,000. What was the amount of the very first payment?

    • A) Rs. 2,000
    • B) Rs. 4,000
    • C) Rs. 6,000
    • D) Rs. 8,000
    Show answer & explanation

    Answer: B) Rs. 4,000

    We have two equations: a + 7d = 18000, and a + 11d = 26000. Subtracting the first from the second gives 4d = 8000, so d = 2000. Substituting d into the first equation: a + 7(2000) = 18000 -> a + 14000 = 18000 -> a = 4000.

24. Question 24

    Using the standard mathematical shortcut formula for arithmetic series, what is the exact sum of the first 20 natural numbers (1 + 2 + 3 + ... + 20)?

    • A) 190
    • B) 200
    • C) 210
    • D) 400
    Show answer & explanation

    Answer: C) 210

    The specific formula for the sum of the first 'n' natural numbers is n(n + 1) / 2. Here, 20(21) / 2 = 420 / 2 = 210.

25. Question 25

    What is the exact Arithmetic Mean inserted between the numbers 15 and 45?

    • A) 25
    • B) 30
    • C) 35
    • D) 60
    Show answer & explanation

    Answer: B) 30

    The Arithmetic Mean (A.M.) between any two numerical values 'a' and 'b' is simply their average, calculated as (a + b) / 2. Here, (15 + 45) / 2 = 60 / 2 = 30.

26. Question 26

    What is the positive Geometric Mean inserted precisely between the numbers 9 and 25?

    • A) 15
    • B) 17
    • C) 34
    • D) 225
    Show answer & explanation

    Answer: A) 15

    The Geometric Mean (G.M.) between two positive numbers 'a' and 'b' is calculated mathematically as the square root of their product: √(a * b). Here, √(9 * 25) = √225 = 15.

27. Question 27

    If the numbers 2, x, and 50 explicitly form a geometric progression, what are the possible mathematical values for 'x'?

    • A) 10 only
    • B) 25 only
    • C) 10 or -10
    • D) 25 or -25
    Show answer & explanation

    Answer: C) 10 or -10

    For three terms to form a geometric progression, the ratio must be constant: x/2 = 50/x. Cross-multiplying gives x^2 = 100. Taking the square root gives x = 10 or x = -10.

28. Question 28

    In any given arithmetic progression, how is the 'common difference' strictly derived?

    • A) By dividing any term by the term that precedes it.
    • B) By subtracting any term from the term that strictly follows it.
    • C) By adding the first and last term together.
    • D) By taking the square root of the series.
    Show answer & explanation

    Answer: B) By subtracting any term from the term that strictly follows it.

    The common difference (d) is defined as the constant value added to each term to get the next. Mathematically, it is found by taking any term (a_n) and subtracting the previous term (a_{n-1}).

29. Question 29

    A worker saves Rs. 500 in month 1 and commits to increasing the savings by Rs. 200 every single month thereafter. What is the total cumulative amount saved after exactly 12 months?

    • A) Rs. 16,400
    • B) Rs. 18,200
    • C) Rs. 19,200
    • D) Rs. 20,400
    Show answer & explanation

    Answer: C) Rs. 19,200

    This forms an arithmetic series where a = 500, d = 200, and n = 12. Sum = (n/2)[2a + (n-1)d] = (12/2)[1000 + (11)200] = 6(1000 + 2200) = 6(3200) = Rs. 19,200.

30. Question 30

    Under what strict mathematical condition does an infinite geometric progression possess a calculable, finite sum?

    • A) Only when the common ratio 'r' is exactly 1.
    • B) When the common ratio 'r' is strictly greater than 1.
    • C) When the absolute value of the common ratio 'r' is strictly less than 1 (-1 < r < 1).
    • D) When the first term is a negative number.
    Show answer & explanation

    Answer: C) When the absolute value of the common ratio 'r' is strictly less than 1 (-1 < r < 1).

    A geometric series converges (approaches a finite sum) as n approaches infinity only if its common ratio 'r' is a fraction between -1 and 1. Otherwise, the sum grows infinitely large.

31. Question 31

    In an Arithmetic Progression (AP), the difference between any two consecutive terms is always:

    • A) Increasing
    • B) Decreasing
    • C) Constant
    • D) Zero
    Show answer & explanation

    Answer: C) Constant

    An AP is defined by its 'common difference' (d), which remains identical between any two neighbors in the sequence (e.g., 2, 5, 8, 11...).

32. Question 32

    What is the standard formula for the n-th term (Tn) of an Arithmetic Progression?

    • A) Tn = a × r^(n-1)
    • B) Tn = a + (n - 1)d
    • C) Tn = (n/2)[2a + (n-1)d]
    • D) Tn = a + nd
    Show answer & explanation

    Answer: B) Tn = a + (n - 1)d

    To find any term in an AP, you start with the first term 'a' and add the common difference 'd' precisely (n-1) times.

33. Question 33

    In a Geometric Progression (GP), the ratio between any two consecutive terms is always:

    • A) Constant
    • B) Increasing
    • C) Decreasing
    • D) Negative
    Show answer & explanation

    Answer: A) Constant

    A GP is defined by its 'common ratio' (r). Each term is found by multiplying the previous term by the same fixed number (e.g., 2, 4, 8, 16...).

34. Question 34

    What is the standard formula for the n-th term (Tn) of a Geometric Progression?

    • A) Tn = a + (n-1)d
    • B) Tn = a × r^(n-1)
    • C) Tn = a × r^n
    • D) Tn = a / r^(n-1)
    Show answer & explanation

    Answer: B) Tn = a × r^(n-1)

    In a GP, you start with the first term 'a' and multiply it by the common ratio 'r' precisely (n-1) times.

35. Question 35

    The sum of the first 10 terms of an AP starting with 2 and having a common difference of 3 is:

    • A) 150
    • B) 155
    • C) 160
    • D) 165
    Show answer & explanation

    Answer: B) 155

    Formula: Sn = (n/2)[2a + (n-1)d]. Here, S10 = (10/2)[2(2) + (9)(3)] = 5[4 + 27] = 5[31] = 155.

36. Question 36

    Which of the following describes a 'Convergent' Geometric Series?

    • A) A GP where the common ratio r is greater than 1.
    • B) A GP where the common ratio r is perfectly between -1 and 1.
    • C) An AP where the common difference is zero.
    • D) Any GP that has more than 1,000 terms.
    Show answer & explanation

    Answer: B) A GP where the common ratio r is perfectly between -1 and 1.

    Only when the absolute value of 'r' is less than 1 will the terms get smaller and smaller, allowing the sum to 'converge' to a specific finite number.

37. Question 37

    What is the formula for the 'Sum to Infinity' (S∞) of a convergent GP?

    • A) S∞ = a / (1 - r)
    • B) S∞ = a / (r - 1)
    • C) S∞ = a(1 - r^n) / (1 - r)
    • D) S∞ = a × r
    Show answer & explanation

    Answer: A) S∞ = a / (1 - r)

    This shortcut formula is extremely useful in finance for calculating the present value of perpetuities.

38. Question 38

    If the 1st term of an AP is 5 and the 10th term is 32, what is the common difference (d)?

    • A) 2
    • B) 3
    • C) 4
    • D) 5
    Show answer & explanation

    Answer: B) 3

    Use Tn = a + (n-1)d. 32 = 5 + (9)d -> 27 = 9d -> d = 3.

39. Question 39

    What is the common ratio (r) of the GP: 100, 50, 25...?

    • A) 2
    • B) -2
    • C) 0.5
    • D) -0.5
    Show answer & explanation

    Answer: C) 0.5

    Ratio = 2nd term / 1st term. 50 / 100 = 0.5 (or 1/2).

40. Question 40

    The sequence 5, 10, 20, 40... is an example of a/an:

    • A) Arithmetic progression
    • B) Geometric progression
    • C) Harmonic progression
    • D) Linear function
    Show answer & explanation

    Answer: B) Geometric progression

    Each term is double the previous term, meaning there is a constant common ratio of 2.

41. Question 41

    In the GP formula for the sum of 'n' terms, if r > 1, the formula used is:

    • A) a(r^n - 1) / (r - 1)
    • B) a(1 - r^n) / (1 - r)
    • C) a / (1 - r)
    • D) n/2 [a + l]
    Show answer & explanation

    Answer: A) a(r^n - 1) / (r - 1)

    While both formulas (A and B) are mathematically equivalent, formula A is preferred when r > 1 to work with positive numbers in the numerator and denominator.

42. Question 42

    What is the sum of the first 5 terms of the GP: 3, 6, 12...?

    • A) 90
    • B) 93
    • C) 96
    • D) 100
    Show answer & explanation

    Answer: B) 93

    a=3, r=2, n=5. Sum = 3(2^5 - 1) / (2 - 1) = 3(32 - 1) / 1 = 3 × 31 = 93.

43. Question 43

    The 'Arithmetic Mean' (AM) between two numbers 'x' and 'y' is defined as:

    • A) √(xy)
    • B) (x + y) / 2
    • C) x × y
    • D) (x - y) / 2
    Show answer & explanation

    Answer: B) (x + y) / 2

    By definition, the arithmetic mean is the simple average of two values.

44. Question 44

    The 'Geometric Mean' (GM) between two positive numbers 'x' and 'y' is defined as:

    • A) (x + y) / 2
    • B) √(xy)
    • C) x^2 + y^2
    • D) x / y
    Show answer & explanation

    Answer: B) √(xy)

    If three numbers x, G, y form a GP, then G/x = y/G, resulting in G^2 = xy, or G = √(xy).

45. Question 45

    If you add 4 to each term of an AP, the new sequence will:

    • A) Become a GP.
    • B) Stay an AP with the same common difference.
    • C) Stay an AP but the common difference will increase by 4.
    • D) Become random.
    Show answer & explanation

    Answer: B) Stay an AP with the same common difference.

    Adding a constant to every term shifts the origin but does not change the physical distance (gap) between neighbors.