CSAT· Prelims
Number Series for CSAT
Number series for CSAT: the pattern checklist covering AP, GP, squares, cubes, difference ladders, interleaved and Fibonacci series, with worked examples and practice MCQs.
Number series is a sequence of numbers in which each term is linked to the previous ones by a hidden rule. Your job is to find the rule and use it to supply the missing term or the next term. It matters for CSAT because Paper II is a qualifying paper needing a 33 percent score, and series questions are the purest reasoning questions on it: no formula sheet helps, but a fixed checklist of pattern families cracks nearly every series UPSC sets.
The pattern checklist
Work through the checklist in this order. 1. Differences: subtract each term from the next; if the differences are constant, it is an arithmetic series, and if the second differences are constant, the rule is quadratic. 2. Ratios: divide each term by the previous one; a constant ratio means a geometric series. 3. Squares and cubes: check whether each term is n^2 or n^3 plus or minus a small constant. 4. Primes: check whether the terms or the differences are prime numbers. 5. Digit games: check sums or products of digits. 6. Interleaved series: split the terms into alternate positions; two simpler series may be hiding inside one.
Example 1. Find the next term: 3, 8, 15, 24, 35, ?. The differences are 5, 7, 9, 11, which grow by 2 each time. The next difference is 13, so the next term is 35 + 13 = 48. Equivalently, the terms are n^2 - 1 for n = 2, 3, 4, 5, 6, so the next is 7^2 - 1 = 48.
Arithmetic and geometric series
An arithmetic progression (AP) adds a fixed common difference d to each term: a, a+d, a+2d, .... Its nth term is a + (n-1)d and the sum of the first n terms is n/2 x (first + last). A geometric progression (GP) multiplies each term by a fixed common ratio r: a, ar, ar^2, .... Its nth term is a x r^(n-1). CSAT rarely asks for sums directly, but recognising AP and GP instantly saves the time you need for harder patterns.
Example 2. Find the 10th term of 5, 9, 13, 17, .... This is an AP with a = 5 and d = 4. The 10th term = 5 + 9 x 4 = 5 + 36 = 41.
Example 3. Find the missing term: 3, 6, 12, 24, ?, 96. Each term doubles the previous one, so it is a GP with ratio 2. The missing term is 24 x 2 = 48, and 48 x 2 = 96 confirms it.
Squares, cubes and their neighbours
Many series are built on squares (1, 4, 9, 16, 25, 36, ...) or cubes (1, 8, 27, 64, 125, ...) with a small adjustment. Memorise squares up to 20^2 and cubes up to 10^3; then test term = square plus or minus 1, 2 or 3, or term = cube plus or minus 1 or 2.
Example 4. Find the next term: 2, 5, 10, 17, 26, ?. Compare with squares: 1^2+1 = 2, 2^2+1 = 5, 3^2+1 = 10, 4^2+1 = 17, 5^2+1 = 26. The pattern is n^2 + 1, so the next term is 6^2 + 1 = 37.
Example 5. Find the next term: 1, 8, 27, 64, 125, ?. These are 1^3, 2^3, 3^3, 4^3, 5^3, so the next is 6^3 = 216.
Difference ladders and two-level patterns
A difference ladder is built by writing the differences between consecutive terms, and then the differences of those differences. If the second-level differences are constant, the original series follows a quadratic rule. A related family adds increasing gaps: the differences themselves form a simple series such as multiples of 3 or the odd numbers.
Example 6. Find the next term: 4, 7, 13, 22, 34, ?. First differences: 3, 6, 9, 12. These grow by 3 each time, so the next difference is 15 and the next term is 34 + 15 = 49.
Interleaved and mixed series
An interleaved series weaves two independent series together in alternate positions: odd-positioned terms follow one rule and even-positioned terms follow another. A mixed series applies two operations in turn, such as x2 then +3, or adds the previous two terms like the Fibonacci sequence. Whenever a series looks chaotic, split it into odd and even positions first.
Example 7. Find the next term: 2, 12, 4, 24, 8, 48, ?. Odd positions: 2, 4, 8 (doubling). Even positions: 12, 24, 48 (doubling). The next term is in an odd position, so it is 8 x 2 = 16.
Example 8. Find the next term: 1, 1, 2, 3, 5, 8, ?. Each term is the sum of the previous two (Fibonacci), so the next is 5 + 8 = 13.
Common traps and speed tips
- Trap 1: assuming one level only. If the first differences look random, write the second differences before giving up.
- Trap 2: ignoring interleaving. A series that jumps up and down (2, 12, 4, 24, 8, 48) is almost always two series woven together.
- Trap 3: prime traps. Series like 2, 3, 5, 7, 11, 13 are just primes; do not force an arithmetic rule onto them.
- Tip 1: memorise the base sequences. Squares to 20^2, cubes to 10^3, powers of 2 to 2^10, and primes to 50 cover most series.
- Tip 2: check +1/-1 around squares and cubes first. n^2 +/- 1 and n^3 +/- 1 are the most common 'neighbour' patterns in CSAT.
Find the missing term: 7, 14, 28, 56, ?, 224.
Show answer
Answer: (B) Each term doubles: 7 x 2 = 14, 14 x 2 = 28, 28 x 2 = 56, 56 x 2 = 112, 112 x 2 = 224.
Find the next term: 121, 144, 169, 196, ?.
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Answer: (B) These are squares: 11^2 = 121, 12^2 = 144, 13^2 = 169, 14^2 = 196, so the next is 15^2 = 225.
Find the missing term: 5, 11, 23, 41, 65, ?.
Show answer
Answer: (B) Differences are 6, 12, 18, 24 (multiples of 6); the next difference is 30, so the next term is 65 + 30 = 95.
Find the next term: 3, 15, 35, 63, 99, ?.
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Answer: (B) Each term is one less than an even square: 2^2 - 1 = 3, 4^2 - 1 = 15, 6^2 - 1 = 35, 8^2 - 1 = 63, 10^2 - 1 = 99, so the next is 12^2 - 1 = 143.
Find the missing term: 2, 6, 12, 20, 30, ?.
Show answer
Answer: (B) Differences are 4, 6, 8, 10 (even numbers); the next difference is 12, so the next term is 30 + 12 = 42. (Each term is also n(n+1): 5 x 6 = 30, 6 x 7 = 42.)
Find the next term: 1, 4, 9, 16, 25, 36, ?.
Show answer
Answer: (C) These are consecutive squares 1^2 to 6^2, so the next is 7^2 = 49.
Find the next term: 5, 11, 23, 47, 95, ?.
Show answer
Answer: (C) Each term is double the previous plus 1: 5 x 2 + 1 = 11, 11 x 2 + 1 = 23, 47 x 2 + 1 = 95, so the next is 95 x 2 + 1 = 191.
Answer key
- Q1 - (b). Each term doubles: 7 x 2 = 14, 14 x 2 = 28, 28 x 2 = 56, 56 x 2 = 112, 112 x 2 = 224.
- Q2 - (b). These are squares: 11^2 = 121, 12^2 = 144, 13^2 = 169, 14^2 = 196, so the next is 15^2 = 225.
- Q3 - (b). Differences are 6, 12, 18, 24 (multiples of 6); the next difference is 30, so the next term is 65 + 30 = 95.
- Q4 - (b). Each term is one less than an even square: 2^2 - 1 = 3, 4^2 - 1 = 15, 6^2 - 1 = 35, 8^2 - 1 = 63, 10^2 - 1 = 99, so the next is 12^2 - 1 = 143.
- Q5 - (b). Differences are 4, 6, 8, 10 (even numbers); the next difference is 12, so the next term is 30 + 12 = 42. (Each term is also n(n+1): 5 x 6 = 30, 6 x 7 = 42.)
- Q6 - (c). These are consecutive squares 1^2 to 6^2, so the next is 7^2 = 49.
- Q7 - (c). Each term is double the previous plus 1: 5 x 2 + 1 = 11, 11 x 2 + 1 = 23, 47 x 2 + 1 = 95, so the next is 95 x 2 + 1 = 191.
What should I check first in any series?
The differences between consecutive terms. A constant difference means an arithmetic progression, and that one check resolves the series instantly. If the differences grow steadily, look at the second differences or at how the differences themselves are changing.
How do I spot an interleaved series?
Look for a zigzag: the series rises and falls without a clear single rule, like 2, 12, 4, 24, 8, 48. Write the odd-positioned terms (2, 4, 8) and even-positioned terms (12, 24, 48) separately; each half usually follows a simple doubling or addition rule.
Are Fibonacci questions common in CSAT?
Yes, in gentle forms: 1, 1, 2, 3, 5, 8, 13 or variants where each term is the sum of the previous two plus a constant. Always test 'term = sum of previous two' when differences look like they are growing fast.
What is the fastest way to learn the base sequences?
Write out squares from 1^2 to 20^2, cubes from 1^3 to 10^3, powers of 2 from 2^1 to 2^10, and primes up to 50, and revise them weekly. Most CSAT series are one of these with a small twist like plus or minus 1.