Number Series Questions and Answers

Q: Select the number from among the given options that can replace the question mark (?) in the following series. 8, 32, 96, 384, ?, 4608

The missing term is 1152.The terms are multiplied alternately by 4 and 3: 8 × 4 = 32, 32 × 3 = 96, 96 × 4 = 384, 384 × 3 = 1152 and 1152 × 4 = 4608, which matches the last term given.960, 1536 and 1056 do not lead to 4608 by the alternate factors 4 and 3.You can confirm the pattern by dividing neighbouring terms: 32 / 8 = 4, 96 / 32 = 3, 384 / 96 = 4, so the multipliers alternate between 4 and 3. The term after 384 is found by multiplying by 3, and the last step to 4608 multiplies by 4 again, which confirms 1152.

Q: In the sequence 6, 9, 14, x, 30, 41, a possible value of \(x\) is

A possible value is 21.The differences between consecutive terms are consecutive odd numbers: 9 - 6 = 3 and 14 - 9 = 5, so the next differences should be 7, 9 and 11. That gives \(14 + 7 = 21\), then \(21 + 9 = 30\) and \(30 + 11 = 41\), matching the terms given after x.With 25, 18 or 20 the differences on either side of x (11 and 5, 4 and 12, 6 and 10) break the pattern, and none of them leads to 30 and 41 by a consistent rule.

Q: Which one of the following options is correct for the given data in the table?Iteration (i)0123Input (I)20-41015Output (X)20162641Output (Y)20-80-800-12000

The relations are X(i) = X(i - 1) + I(i) and Y(i) = Y(i - 1) I(i), for i > 0.Check X: 20 + (-4) = 16, 16 + 10 = 26, 26 + 15 = 41. Each output adds the new input to the previous output.Check Y: 20 × (-4) = -80, -80 × 10 = -800, -800 × 15 = -12000. Each output multiplies the previous output by the new input.The option with X multiplying would give 20 × (-4) = -80 for X(1), not 16, and Y adding would give 16, not -80. The last option multiplies Y by the previous input, giving Y(1) = 20 × 20 = 400 instead of -80.

Q: What is the missing number in the following sequence?2, 12, 60, 240, 720, 1440, _____, 0

The missing number is 1440.Each term is the previous one multiplied by a factor that falls by one each time: 2 × 6 = 12, 12 × 5 = 60, 60 × 4 = 240, 240 × 3 = 720, 720 × 2 = 1440. The next factor is 1, giving 1440 × 1 = 1440, and then 0, giving 1440 × 0 = 0, which matches the last term.2880 continues doubling, which is only the last step's factor. 720 and 0 do not fit the falling multipliers, and 0 is reserved for the final term, which needs the factor 0.

Q: A series of natural numbers \(F_{1}, F_{2}, F_{3}, F_{4}, F_{5}, F_{6}, F_{7}, \ldots\) obeys \(F_{n+1} = F_{n} + F_{n-1}\) for all integers \(n \ge 2\).If \(F_{6} = 37\), and \(F_{7} = 60\), then what is \(F_{1}\)?

\(F_{1}\) is 4.The rule \(F_{n+1} = F_{n} + F_{n-1}\) can be run backwards as \(F_{n-1} = F_{n+1} - F_{n}\):\[ \begin{aligned} F_{5} &= 60 - 37 = 23 \\ F_{4} &= 37 - 23 = 14 \\ F_{3} &= 23 - 14 = 9 \\ F_{2} &= 14 - 9 = 5 \\ F_{1} &= 9 - 5 = 4 \end{aligned} \]Check forwards: 4, 5, 9, 14, 23, 37, 60. The value 5 is \(F_{2}\), 9 is \(F_{3}\), and 8 does not appear in the series at all.

Q: In the following series, identify the number that needs to be changed to form the Fibonacci series.1, 1, 2, 3, 6, 8, 13, 21,...

The number to change is 6.In the Fibonacci series each term is the sum of the two before it: 1, 1, 2, 3, 5, 8, 13, 21, ... Here the fifth term should be \(2 + 3 = 5\), not 6.With 5 in place, every later term checks: \(3 + 5 = 8\), \(5 + 8 = 13\), \(8 + 13 = 21\). So 8, 13 and 21 are all correct as printed; it is only the 6 that breaks the rule.

Q: In the given series, one of the term is wrong. Find that one.2, 3, 8, 63, 3962.

The wrong term is 3962.Each term is the square of the one before it, minus 1:\[ \begin{aligned} 2^2 - 1 &= 3 \\ 3^2 - 1 &= 8 \\ 8^2 - 1 &= 63 \\ 63^2 - 1 &= 3968 \end{aligned} \]The last term should be 3968, so 3962 is the one that breaks the rule. 3, 8 and 63 all fit the pattern exactly, so none of them is the wrong term. A quick check without full multiplication: \(63^2\) ends in 9, so \(63^2 - 1\) must end in 8, not 2.

Q: Given two identical sequences: [4000, 400.0, 120.0, 60.0, 42.0, 37.8] and [1000, A, B, C, D, E]. What is the entry at the place of C for the second sequence?

The entry at C is 15.0.Find the rule of the first sequence from the ratio of each term to the one before:\[ \begin{aligned} 400/4000 &= 0.1,\ 120/400 = 0.3 \\ 60/120 &= 0.5,\ 42/60 = 0.7 \\ 37.8/42 &= 0.9 \end{aligned} \]The multipliers are 0.1, 0.3, 0.5, 0.7, 0.9. Applying the same rule from 1000: A = 100, B = 30, C = 15, D = 10.5, E = 9.45. So 10.5 and 9.45 are the entries at D and E, and 30.0 is the entry at B; they are the off-by-one traps.

Important Instructions

171.

Select the number from among the given options that can replace the question mark (?) in the following series. 3, 50, 755, 11330 ?

A.169955
B.169950
C.169590
D.169055

Exam Year(s)

DSSSB PGT - 2023,

Answer: A

The missing number is 169955. Each term is 15 times the previous one plus 5: 3 x 15 + 5 = 50, 50 x 15 + 5 = 755, and 755 x 15 + 5 = 11330. Applying the rule once more gives 11330 x 15 + 5 = 169950 + 5 = 169955. The values 169950, 169590 and 169055 do not equal 11330 x 15 + 5. The same multiplier can be checked backwards: 11330 minus 5 is 11325, and 11325 divided by 15 is 755; likewise 755 minus 5 is 750, and 750 divided by 15 is 50. Because the rule reproduces every given term, applying it forward is safe.

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172.

Select the number from among the given options that can replace the question mark (?) in the following series. 2, 6, 24, 120, 720, ?

A.4050
B.5040
C.5400
D.5004

Exam Year(s)

DSSSB - 2023,

Answer: B

The series 2, 6, 24, 120, 720 is the series of factorials: 2 = 2!, 6 = 3!, 24 = 4!, 120 = 5!, 720 = 6!. Each term is the previous term multiplied by the next whole number, 3, 4, 5, 6. The next term is 720 x 7 = 5040, which is 7!.

\[ 720 \times 7 = 5040 \]

The options 4050, 5400 and 5004 are not 7 times 720, so they cannot be the next term of the series.

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173.

Select the number from among the given options that can replace the question mark (?) in the following series. 8, 32, 96, 384, ?, 4608

A.960
B.1152
C.1536
D.1056

Exam Year(s)

DSSSB - 2023,

Answer: B

The missing term is 1152.

The terms are multiplied alternately by 4 and 3: 8 × 4 = 32, 32 × 3 = 96, 96 × 4 = 384, 384 × 3 = 1152 and 1152 × 4 = 4608, which matches the last term given.

960, 1536 and 1056 do not lead to 4608 by the alternate factors 4 and 3.

You can confirm the pattern by dividing neighbouring terms: 32 / 8 = 4, 96 / 32 = 3, 384 / 96 = 4, so the multipliers alternate between 4 and 3. The term after 384 is found by multiplying by 3, and the last step to 4608 multiplies by 4 again, which confirms 1152.

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174.

In the sequence 6, 9, 14, x, 30, 41, a possible value of \(x\) is

A.25
B.21
C.18
D.20

Exam Year(s)

GATE Life Sciences - 2024, GATE CS - 2024, GATE XE - 2024,

Answer: B

A possible value is 21.

The differences between consecutive terms are consecutive odd numbers: 9 - 6 = 3 and 14 - 9 = 5, so the next differences should be 7, 9 and 11. That gives \(14 + 7 = 21\), then \(21 + 9 = 30\) and \(30 + 11 = 41\), matching the terms given after x.

With 25, 18 or 20 the differences on either side of x (11 and 5, 4 and 12, 6 and 10) break the pattern, and none of them leads to 30 and 41 by a consistent rule.

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175.

Which one of the following options is correct for the given data in the table?

Iteration (i)	0	1	2	3
Input (I)	20	-4	10	15
Output (X)	20	16	26	41
Output (Y)	20	-80	-800	-12000

A.\(X(i) = X(i-1) + I(i);\ Y(i) = Y(i-1)I(i);\ i \gt 0\)
B.\(X(i) = X(i-1)I(i);\ Y(i) = Y(i-1) + I(i);\ i \gt 0\)
C.\(X(i) = X(i-1)I(i);\ Y(i) = Y(i-1)I(i);\ i \gt 0\)
D.\(X(i) = X(i-1) + I(i);\ Y(i) = Y(i-1)I(i-1);\ i \gt 0\)

Exam Year(s)

GATE CS - 2025,

Answer: A

The relations are X(i) = X(i - 1) + I(i) and Y(i) = Y(i - 1) I(i), for i > 0.

Check X: 20 + (-4) = 16, 16 + 10 = 26, 26 + 15 = 41. Each output adds the new input to the previous output.

Check Y: 20 × (-4) = -80, -80 × 10 = -800, -800 × 15 = -12000. Each output multiplies the previous output by the new input.

The option with X multiplying would give 20 × (-4) = -80 for X(1), not 16, and Y adding would give 16, not -80. The last option multiplies Y by the previous input, giving Y(1) = 20 × 20 = 400 instead of -80.

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176.

What is the missing number in the following sequence?

2, 12, 60, 240, 720, 1440, _____, 0

A.2880
B.1440
C.720
D.0

Exam Year(s)

GATE CS - 2018,

Answer: B

The missing number is 1440.

Each term is the previous one multiplied by a factor that falls by one each time: 2 × 6 = 12, 12 × 5 = 60, 60 × 4 = 240, 240 × 3 = 720, 720 × 2 = 1440. The next factor is 1, giving 1440 × 1 = 1440, and then 0, giving 1440 × 0 = 0, which matches the last term.

2880 continues doubling, which is only the last step's factor. 720 and 0 do not fit the falling multipliers, and 0 is reserved for the final term, which needs the factor 0.

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177.

A series of natural numbers \(F_{1}, F_{2}, F_{3}, F_{4}, F_{5}, F_{6}, F_{7}, \ldots\) obeys \(F_{n+1} = F_{n} + F_{n-1}\) for all integers \(n \ge 2\).

If \(F_{6} = 37\), and \(F_{7} = 60\), then what is \(F_{1}\)?

Exam Year(s)

GATE CS - 2023,

Answer: A

\(F_{1}\) is 4.

The rule \(F_{n+1} = F_{n} + F_{n-1}\) can be run backwards as \(F_{n-1} = F_{n+1} - F_{n}\):

\[ \begin{aligned} F_{5} &= 60 - 37 = 23 \\ F_{4} &= 37 - 23 = 14 \\ F_{3} &= 23 - 14 = 9 \\ F_{2} &= 14 - 9 = 5 \\ F_{1} &= 9 - 5 = 4 \end{aligned} \]

Check forwards: 4, 5, 9, 14, 23, 37, 60. The value 5 is \(F_{2}\), 9 is \(F_{3}\), and 8 does not appear in the series at all.

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178.

In the following series, identify the number that needs to be changed to form the Fibonacci series.

1, 1, 2, 3, 6, 8, 13, 21,...

A.8
B.21
C.6
D.13

Exam Year(s)

GATE Physics - 2024, GATE ME - 2024,

Answer: C

The number to change is 6.

In the Fibonacci series each term is the sum of the two before it: 1, 1, 2, 3, 5, 8, 13, 21, ... Here the fifth term should be \(2 + 3 = 5\), not 6.

With 5 in place, every later term checks: \(3 + 5 = 8\), \(5 + 8 = 13\), \(8 + 13 = 21\). So 8, 13 and 21 are all correct as printed; it is only the 6 that breaks the rule.

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179.

In the given series, one of the term is wrong. Find that one.

2, 3, 8, 63, 3962.

A.3
B.8
C.3962
D.63

Exam Year(s)

HPPSC HPAS - 2021,

Answer: C

The wrong term is 3962.

Each term is the square of the one before it, minus 1:

\[ \begin{aligned} 2^2 - 1 &= 3 \\ 3^2 - 1 &= 8 \\ 8^2 - 1 &= 63 \\ 63^2 - 1 &= 3968 \end{aligned} \]

The last term should be 3968, so 3962 is the one that breaks the rule. 3, 8 and 63 all fit the pattern exactly, so none of them is the wrong term. A quick check without full multiplication: \(63^2\) ends in 9, so \(63^2 - 1\) must end in 8, not 2.

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180.

Given two identical sequences: [4000, 400.0, 120.0, 60.0, 42.0, 37.8] and [1000, A, B, C, D, E]. What is the entry at the place of C for the second sequence?

A.9.45
B.10.5
C.15.0
D.30.0

Exam Year(s)

Haryana Civil Services - 2026, HPSC - 2026,

Answer: C

The entry at C is 15.0.

Find the rule of the first sequence from the ratio of each term to the one before:

\[ \begin{aligned} 400/4000 &= 0.1,\ 120/400 = 0.3 \\ 60/120 &= 0.5,\ 42/60 = 0.7 \\ 37.8/42 &= 0.9 \end{aligned} \]

The multipliers are 0.1, 0.3, 0.5, 0.7, 0.9. Applying the same rule from 1000: A = 100, B = 30, C = 15, D = 10.5, E = 9.45. So 10.5 and 9.45 are the entries at D and E, and 30.0 is the entry at B; they are the off-by-one traps.

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