UPSC CSAT Number Series PYQ with Solutions
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Number Series Previous Questions (2011-2025) Solved
Struggling with Number Series in UPSC CSAT?
This complete guide covers solved PYQs (2011–2025), smart pattern recognition tricks, detailed explanations to boost your score.
Year-wise Analysis of Number Series PYQs (2011–2025)
Q1. What is the missing number ‘X’ of the series: 7, X, 21, 31, 43 ? (CSAT 2015)
(a) 11
(b) 12
(c) 13
(d) 14
Correct Answer: (c) 13
Solution:
Method 1: Difference Approach
Step 1: Analyze the known numbers.
31 − 21 = 10
43 − 31 = 12
Step 2: Identify the pattern.
The differences are increasing even numbers: 10, 12.
So the previous even numbers should be 6 and 8.
Step 3: Apply backward.
21 − 8 = 13
13 − 6 = 7
Since this matches the first number, the logic is correct.
Therefore, the missing number is 13.
Method 2: Squares Approach
The series follows the pattern:
n2 + (n + 1)
22 + 3 = 7
32 + 4 = 13
42 + 5 = 21
52 + 6 = 31
62 + 7 = 43
Hence, the missing number is 13.
Q2. What is X in the sequence 4, 196, 16, 144, 36, 100, 64, X? (CSAT 2019)
(a) 48
(b) 64
(c) 125
(d) 256
Correct Answer: (b) 64
Solution
Method 1: The “Twin Series” Approach
Instead of looking at the numbers in a straight line, imagine two separate chains interlaced with each other:
the Odd positions (1st, 3rd, 5th…) and the Even positions (2nd, 4th, 6th…).
Chain A (Odd Positions): Increasing Squares
4 → (22)
16 → (42)
36 → (62)
64 → (82)
Logic: Squares of increasing even numbers (2, 4, 6, 8…)
Chain B (Even Positions): Decreasing Squares
196 → (142)
144 → (122)
100 → (102)
X → ( ? )
Logic: Squares of decreasing even numbers (14, 12, 10…)
Following the pattern of Chain B, the number after 10 must be 8.
Therefore,
X = 82 = 64
Method 2: The Constant Sum Approach
If you prefer algebra or spotting relationships between pairs, this method is often faster for this type of question.
Group the numbers into pairs (1st & 2nd, 3rd & 4th, etc.) and observe their square roots.
The Logic:
The sum of the square roots of each pair is always 16.
Pair 1 (4, 196):
√4 + √196 = 2 + 14 = 16
Pair 2 (16, 144):
√16 + √144 = 4 + 12 = 16
Pair 3 (36, 100):
√36 + √100 = 6 + 10 = 16
Pair 4 (64, X):
√64 + √X = 16
8 + √X = 16
√X = 8
X = 82 = 64
Q3. What is X in the sequence 132, 129, 124, 117, 106, 93, X? (CSAT 2019)
(a) 74
(b) 75
(c) 76
(d) 77
Correct Answer: (c) 76
Solution
Method 1: The “Prime Drop”
The key to this series is identifying that the numbers are decreasing by consecutive
Prime Numbers, not just odd numbers.
1. Calculate the gaps:
132 − 129 = 3
129 − 124 = 5
124 − 117 = 7
117 − 106 = 11
106 − 93 = 13
2. Analyze the gaps:
The difference series becomes:
3, 5, 7, 11, 13
Observation: If this were a simple odd number pattern, the number 9 would appear
between 7 and 11.
Since 9 is missing, this confirms the pattern follows
prime numbers (numbers divisible only by 1 and themselves).
3. Find the next term:
Prime number sequence:
2, 3, 5, 7, 11, 13, 17, 19…
The next prime after 13 is 17.
Subtract it from the last given number:
93 − 17 = 76
Therefore, the next number in the series is 76.
Method 2: The “Pattern Elimination” Approach
In exams like CSAT, you can use the “difference of differences” idea to quickly
identify whether a series is simple or irregular.
Step 1: First layer of differences
3, 5, 7, 11, 13
Step 2: Second layer of differences
5 − 3 = 2
7 − 5 = 2
11 − 7 = 4
13 − 11 = 2
The logic:
If the pattern were purely odd numbers, the gap would always be 2.
The sudden jump from 2 to 4 clearly shows the pattern is not linear.
This irregular gap is a strong indicator of prime number logic.
Once you notice this jump at 11, you can immediately conclude that the next
deduction must be the next prime number (17) and not the next odd number (15).
Q4. A simple mathematical operation in each number of the sequence 14, 18, 20, 24, 30, 32, results in a … sequence with respect to prime number. Which one of the following is the next number in the sequence? ? (CSAT 2020)
(a) 34
(b) 36
(c) 38
(d) 40
Correct Answer: (c) 38
Solution
Method 1: The “Hidden Prime” Pattern
The series is built by taking consecutive prime numbers and adding 1 to them.
The “simple mathematical operation” mentioned is just subtracting 1 to reveal the prime.
1. Decode the pattern (reveal the primes):
14 − 1 = 13
18 − 1 = 17
20 − 1 = 19
24 − 1 = 23
30 − 1 = 29
32 − 1 = 31
2. Analyze the hidden sequence:
The obtained numbers are:
13, 17, 19, 23, 29, 31
These are consecutive prime numbers arranged in correct order, with none skipped.
3. Find the next step:
The prime number immediately after 31 is 37.
Reverse the operation by adding 1 back:
37 + 1 = 38
Therefore, the next number in the series is 38.
[Image: Number series flow chart showing 14, 18, 20, 24, 30, 32 with arrows pointing
down to 13, 17, 19, 23, 29, 31]
Method 2: The “Option Elimination” Strategy
In competitive exams like CSAT, checking the options is often faster than solving the
logic from scratch, especially when the rule is hinted in the question.
The strategy:
If the pattern is (Prime + 1), then (Answer − 1) must be a prime number.
Test each option:
(a) 34 → 34 − 1 = 33
33 is not prime (divisible by 3 and 11). Eliminate.
(b) 36 → 36 − 1 = 35
35 is not prime (divisible by 5 and 7). Eliminate.
(c) 38 → 38 − 1 = 37
37 is prime and comes immediately after 31. Correct.
(d) 40 → 40 − 1 = 39
39 is not prime (divisible by 3 and 13). Eliminate.
By simple primality checking, option (c) 38 is the unique correct answer.
Q5. What is the value of ‘X’ in the sequence 2, 7, 22, 67, 202, X, 1822? (CSAT 2021)
(a) 603
(b) 605
(c) 607
(d) 608
Correct Answer: (c) 607
Solution
Method 1: The “Multiplier” Approach
When a series starts very small (2) and ends very large (1822), it is usually a
multiplication-based series, not simple addition.
1. Estimate the growth:
Look at the numbers: 2, 7, 22…
2 → 7 is roughly ×3 (2 × 3 = 6)
7 → 22 is roughly ×3 (7 × 3 = 21)
22 → 67 is roughly ×3 (about 20 × 3 = 60)
This strongly suggests multiplication by 3.
2. Find the exact rule:
2 × 3 = 6 → need +1 to reach 7
7 × 3 = 21 → need +1 to reach 22
So the pattern becomes:
× 3 + 1
3. Calculate X:
Apply the rule to the number before X, which is 202:
202 × 3 + 1 = 606 + 1 = 607
Self-check:
607 × 3 + 1 = 1821 + 1 = 1822
The logic is confirmed.
[Image: Number series flow chart showing 2, 7, 22, 67, 202, X, 1822 with arrows
indicating ×3 + 1]
Method 2: The “Gap Expansion” Approach
If multiplication logic is difficult to notice, the same series can be solved
by observing the gaps between consecutive numbers.
Step 1: Find the differences:
7 − 2 = 5
22 − 7 = 15
67 − 22 = 45
202 − 67 = 135
Step 2: Analyze the gaps:
Gap sequence:
5, 15, 45, 135
5 × 3 = 15
15 × 3 = 45
45 × 3 = 135
The gaps themselves are multiplied by 3.
Step 3: Find the next gap:
135 × 3 = 405
Step 4: Add the gap:
202 + 405 = 607
Therefore, the value of X = 607.
Q6. Replace the incorrect term by the correct term in the given sequence 3, 2, 7, 4, 13, 10, 21, 18, 31, 28, 43, 40 where odd terms and even terms follow the same pattern. (CSAT 2021)
(a) 0
(b) 1
(c) 3
(d) 6
Correct Answer: (a) 0
Solution
Method 1: The “Parallel Tracks” Approach
The problem states that odd terms and even terms follow the same pattern.
This means the sequence must be treated as two separate tracks,
and their gap rules must match exactly.
Track A (Odd Positions):
3, 7, 13, 21, 31, 43
7 − 3 = 4
13 − 7 = 6
21 − 13 = 8
31 − 21 = 10
43 − 31 = 12
Pattern: Gaps are increasing even numbers:
4, 6, 8, 10, 12
Track B (Even Positions):
2, 4, 10, 18, 28, 40
Check the gaps from right to left:
40 − 28 = 12 (matches Track A)
28 − 18 = 10 (matches Track A)
18 − 10 = 8 (matches Track A)
10 − 4 = 6 (matches Track A)
Problem area:
4 − 2 = 2 (does NOT match Track A)
The fix:
To match Track A, the first gap in Track B must be 4, not 2.
We need a number X such that:
4 − X = 4
X = 4 − 4 = 0
Method 2: The “Partner Difference” Shortcut
In many odd–even alternating series, there is a direct relationship between
each odd term and the even term immediately following it.
The logic:
Observe the difference between each odd term and its neighboring even term.
Pair 2: 7 − 4 = 3
Pair 3: 13 − 10 = 3
Pair 4: 21 − 18 = 3
Pair 5: 31 − 28 = 3
Pair 6: 43 − 40 = 3
Pattern: Each pair has a constant difference of 3.
The anomaly:
Pair 1: (3, 2)
3 − 2 = 1
This breaks the constant difference rule.
The solution:
To maintain the constant difference of 3, keep the odd term fixed and adjust the even term:
3 − X = 3
X = 0
Therefore, the missing number is 0.
Q7. What is the value of X in the sequence 20, 10, 10, 15, 30, 75, X? (CSAT 2022)
(a) 105
(b) 120
(c) 150
(d) 225
Correct Answer: (d) 225
Solution
Method 1: The “Half-Step” Pattern
The biggest clue in this series is the beginning:
20 → 10 → 10.
Whenever a series drops by half and then repeats the same number, the pattern is
almost always:
× 0.5, then × 1
1. Trace the multipliers:
20 × 0.5 = 10 (half)
10 × 1.0 = 10 (same)
10 × 1.5 = 15 (one and a half)
15 × 2.0 = 30 (double)
30 × 2.5 = 75 (two and a half)
The multipliers are increasing by 0.5 each step.
2. Predict the next step:
Multiplier pattern:
0.5, 1.0, 1.5, 2.0, 2.5
The next multiplier must be 3.0.
3. Calculate:
75 × 3 = 225
[Image: Number series showing 20, 10, 10, 15, 30, 75 with multiplier logic]
Method 2: The “Fractional Ratio” Approach
If working with decimals feels uncomfortable, the same logic can be expressed
using fractions.
The logic:
Take the ratio of the current term to the previous term:
(Current ÷ Previous)
10 ÷ 20 = 1/2
10 ÷ 10 = 1 = 2/2
15 ÷ 10 = 1.5 = 3/2
30 ÷ 15 = 2 = 4/2
75 ÷ 30 = 2.5 = 5/2
The multipliers form a clear sequence:
1/2, 2/2, 3/2, 4/2, 5/2
Find the next term:
The next fraction must be 6/2, which equals 3.
75 × 3 = 225
Therefore, the required answer is 225.
Q8. What is the value of X in the sequence 2, 12, 36, 80, 150, Х? (CSAT 2022)
(a) 248
(b) 252
(c) 258
(d) 262
Correct Answer: (b) 252
Solution
Method 1: The “Sum of Powers” Approach
The series is formed by adding the square and cube of the same number.
1. Decode the pattern:
1st term: 12 + 13 = 1 + 1 = 2
2nd term: 22 + 23 = 4 + 8 = 12
3rd term: 32 + 33 = 9 + 27 = 36
4th term: 42 + 43 = 16 + 64 = 80
5th term: 52 + 53 = 25 + 125 = 150
2. Predict the next term:
The pattern follows natural numbers: 1, 2, 3, 4, 5…
The next number is 6.
3. Calculate X:
62 + 63
36 + 216 = 252
Method 2: The “Triple Difference” Approach
If the power pattern is not immediately visible, the series can always be solved
using the difference method. This works well for polynomial-type series.
Step 1: First layer of differences
12 − 2 = 10
36 − 12 = 24
80 − 36 = 44
150 − 80 = 70
Step 2: Second layer of differences
24 − 10 = 14
44 − 24 = 20
70 − 44 = 26
Step 3: Third layer of differences
20 − 14 = 6
26 − 20 = 6
A constant difference of 6 confirms a cubic-based pattern.
Step 4: Build the series back up
Third layer: next difference = 6
Second layer: 26 + 6 = 32
First layer: 70 + 32 = 102
Final term: 150 + 102 = 252
Therefore, the missing number is 252.
Q9. Choose the group which is different from the others: (CSAT 2023)
(a) 17, 37, 47, 97
(b) 31, 41, 53, 67
(c) 71, 73, 79, 83
(d) 83, 89, 91, 97
Correct Answer: (d) 83, 89, 91, 97
Solution
Method 1: The “Prime Audit”
We need to check whether every number in a group is a prime number.
If even one number can be divided by another number, that group becomes the
different one.
Analyze the groups:
Group (a): 17, 37, 47, 97
All numbers are prime.
Group (b): 31, 41, 53, 67
All numbers are prime.
Group (c): 71, 73, 79, 83
All numbers are prime.
Group (d): 83, 89, 91, 97
83, 89, and 97 are prime numbers.
91 is not a prime number. It is a composite number.
91 = 7 × 13
Since Group (d) contains a non-prime number (91), it is the odd one out.
Method 2: The “Fake Prime” Detection
In exams, verifying every number takes time. Instead, look for
fake primes.
These numbers appear prime because they are odd and do not end in 5,
but they are actually composite.
The strategy:
Common tricky composite numbers in CSAT include:
51 = 17 × 3
57 = 19 × 3
87 = 29 × 3
91 = 13 × 7 (very common CSAT trap)
Applying the strategy:
1. Scan the options quickly.
2. Check whether numbers like 51, 57, 87, or 91 appear.
3. Here, 91 appears in option (d).
Quick verification:
Divisible by 3? → 9 + 1 = 10 (No)
Divisible by 7? → 7 × 13 = 91 (Yes)
Since 91 is divisible by 7, option (d) is the correct answer.
Q10. What will come in place of * in the sequence 3, 14, 39, 84, *, 258? (CSAT 2024)
(a) 150
(b) 155
(c) 160
(d) 176
Correct Answer: (b) 155
Solution
Method 1: The “Sum of Powers” Approach
The series follows the formula:
n3 + n2 + n
(Cube + Square + the number itself)
1. Decode the pattern:
1st term: 13 + 12 + 1 = 1 + 1 + 1 = 3
2nd term: 23 + 22 + 2 = 8 + 4 + 2 = 14
3rd term: 33 + 32 + 3 = 27 + 9 + 3 = 39
4th term: 43 + 42 + 4 = 64 + 16 + 4 = 84
2. Predict the next term:
The pattern follows consecutive natural numbers:
1, 2, 3, 4…
So the missing term is the 5th term.
3. Calculate X:
53 + 52 + 5
125 + 25 + 5 = 150 + 5 = 155
Method 2: The “Triple Difference” Approach
If the cubic pattern is not immediately visible, the difference method can always
be used. For series involving cubes, the third layer of differences becomes constant.
Step 1: First layer of differences
14 − 3 = 11
39 − 14 = 25
84 − 39 = 45
Step 2: Second layer of differences
25 − 11 = 14
45 − 25 = 20
Step 3: Third layer (constant)
20 − 14 = 6
The constant difference is 6.
Step 4: Build the series upwards
New second layer: 20 + 6 = 26
New first layer: 45 + 26 = 71
Final term: 84 + 71 = 155
Therefore, the missing number is 155.
