SSC CGL Reasoning · Series

Squares, Cubes and Primes

Some series are a list you already know: squares, cubes, one more or one less than them, the primes or the factorials. Recognising the list is faster than working out any difference.

All Series notes

Why this matters

Twenty-seven questions in the bank, 22 of them from the 2025 shifts: 23 EASY and four MODERATE. Ten are plain squares, cubes or powers, twelve are one step away from a square or cube (such as one more than a square, or n times n + 1), and five are primes, factorials or the sum of odd numbers. Knowing the squares to 30 and the cubes to 15 turns each into a few seconds of work.

Concept 1 of 3: Squares, cubes and powers

If every term is a number you recognise as a square or a cube, the series is that list. The only work is to see which numbers are being squared or cubed: all of them, only the even ones, the odd ones, the primes, or multiples of 11.

Definition

Squares 11 to 30: 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900.
Cubes 1 to 15: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375.
Which numbers are powered: 16, 36, 64 are the squares of 4, 6, 8 (even); 1, 27, 125 are the cubes of 1, 3, 5 (odd); 4, 9, 25, 49 are squares of the primes.
Powers of their own place: 1, 4, 27, 256 are $1^1, 2^2, 3^3, 4^4$ , so next is $5^5 = 3125$ .
Squares and cubes woven together: check the odd places and the even places on their own.

Worked example

Find the next term: 121, 169, 289, 361, ?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 17 Sep 2025, Shift 3 · Q9Moderate

Example 1 · Series · Squares, Cubes and Primes

Complete the series: 1331, 10648, 35937, 85184, ?

Calling the list too early

64 is both

8^2

and

4^3

, and 729 is both

27^2

and

9^3

. Read the neighbouring terms to decide whether the series is squares or cubes.

Drill 9 more on squares, cubes and powers

Concept 2 of 3: One step away from a square or a cube

Terms like 3, 8, 15, 24 are just short of squares, and 2, 6, 12, 20 sit between them. Add or take away 1 from a few terms, or try n times n + 1, and the hidden square or product appears.

Definition

Square plus 1: 2, 5, 10, 17, 26, 37, 50, 65. Square minus 1: 0, 3, 8, 15, 24, 35, 48.
Cube plus 1: 2, 9, 28, 65, 126. Cube minus 1: 0, 7, 26, 63, 124.
n times n + 1: 2, 6, 12, 20, 30, 42, 56. Using only odd n gives 2, 12, 30, 56, 90.
Other mixes in the bank: $n^3 + n^2$ (2, 12, 36, 80), $n^3 - n$ (0, 6, 24, 60), and $n^2 + 1$ and $n^2 - 1$ taking turns (2, 3, 10, 15).
Check with differences: a square plus a fixed number has differences 3, 5, 7, 9, the odd numbers, so the difference route agrees.

Common near-square and near-cube terms

n^2 \pm 1 \qquad n^3 \pm 1 \qquad n(n+1) \qquad n^3 - n = (n-1)\,n\,(n+1)

$n$ the counting number behind each term

Worked example

Find the next term: 9, 28, 65, 126, ?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 23 Sep 2025, Shift 1 · Q17Easy

Example 2 · Series · Squares, Cubes and Primes

What should come at the place of question mark? 3, 8, 15, 24, 35, ?

Plus one or minus one

24 is

5^2 - 1

, while 26 is

5^2 + 1

. Test the first two terms, not just one, before you decide which way the series goes.

Drill 11 more on one step away from a square or a cube

Concept 3 of 3: Primes, factorials and sums

A few series are lists to know by heart: the primes, the factorials, and the sums of the first odd numbers, which are the squares.

Definition

Primes to 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
Factorials: 1, 2, 6, 24, 120, 720, 5040, 40320, 362880. Each is the one before times the next counting number.
Sum of the first n odd numbers is $n^2$ : $1 + 3 + 5 + 7 = 16$ .
Sum of the first n counting numbers is $\frac{n(n+1)}{2}$ .
Testing a prime quickly: a number from 11 to 120 is prime if none of 2, 3, 5 and 7 divides it.

Sums and factorials

1 + 3 + \dots + (2n - 1) = n^2 \qquad 1 + 2 + \dots + n = \frac{n(n+1)}{2} \qquad n! = n \times (n-1)!

$n$ how many numbers are added, or the number whose factorial is taken

Worked example

Find the next term: 31, 37, 41, 43, ?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 14 Sep 2025, Shift 3 · Q5Easy

Example 3 · Series · Squares, Cubes and Primes

Complete the series: 24,120, 720, 5040, 40320,__________?

Numbers that look prime

51 is

3 \times 17

, 57 is

3 \times 19

, 87 is

3 \times 29

and 91 is

7 \times 13

. Add the digits to test for 3 (a digit sum divisible by 3 means the number is too) and try 7 before calling a number prime.

Drill 4 more on primes, factorials and sums

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

One step away from a square or a cube
Common near-square and near-cube terms
$n^2 \pm 1 \qquad n^3 \pm 1 \qquad n(n+1) \qquad n^3 - n = (n-1)\,n\,(n+1)$
Primes, factorials and sums
Sums and factorials
$1 + 3 + \dots + (2n - 1) = n^2 \qquad 1 + 2 + \dots + n = \frac{n(n+1)}{2} \qquad n! = n \times (n-1)!$

Watch out for (3)

Calling the list too early
→ Squares, cubes and powers
Plus one or minus one
→ One step away from a square or a cube
Numbers that look prime
→ Primes, factorials and sums

Test yourself on Series

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