SSC CGL Reasoning · Series

Difference Series

Write the gap between each pair of next-door terms under the series; the gaps are steady, grow by a fixed step, follow a list you know, or take turns between two rules.

All Series notes

Why this matters

Seventy-five questions in the bank, 33 from the 2023 shifts, 22 from the 2024 shifts and 20 from the 2025 shifts: 71 EASY and four MODERATE. In 44 the differences are steady or grow by a fixed step, in 20 the differences follow a rule of their own (primes, squares, doubling, or a second row of differences), and in eleven two rules take turns. Writing the differences is the first thing to try on any number series, so this page comes before the others.

Concept 1 of 3: Steady and evenly growing differences

Most number series in SSC are built by adding. Write the differences, and you usually see the same number every time, or a row like 4, 7, 10, 13 that grows by the same amount.

Definition

The order a strong student tries: differences first; if the terms grow very fast, ratios next (the Multiplication page); then squares, cubes and primes; then alternate terms.
Same difference: add it again. 245, 212, 179 takes away 33 each time.
Differences that grow by a fixed step: 5, 8, 11 grows by 3, so the next difference is 14.
Missing term in the middle: find the differences on both sides of the gap; the blank must make both of them fit.
Fast route: after you get an answer, add the next difference to it and check that you land on the term printed after it. Two options rarely survive that check.

Next term when the difference grows by a fixed step

t_{n+1} = t_n + d_n \qquad d_{n+1} = d_n + k

$t_n$ the term in place n
$d_n$ the difference after that term
$k$ the fixed amount the difference grows by (0 for a steady series)

Worked example

Find the missing term: 14, 19, 27, 38, ?, 69

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 23 Sep 2025, Shift 3 · Q4Easy

Example 1 · Series · Difference Series

What should come in place of the question mark '?' in the following number series? 9, 14, 24, 39, 59, ?, 114

Not checking the term after the blank

An answer that fits the differences before a middle blank can still break the one after it. Always add the next difference and land on the printed term.

Mixing up the sign of the difference

In a falling series the differences are negative. 70, 58, 47 has differences

-12

and

-11

, so the next is

-10

and the term is 37, not 57.

Drill 43 more on steady and evenly growing differences

Concept 2 of 3: Differences that follow their own rule

Sometimes the row of differences is itself a series you know: the primes, the squares, the doubling numbers. If not, take the differences of the differences, and that second row is usually steady.

Definition

Known lists to recognise in the differences:
Primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.
Squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. Cubes: 1, 8, 27, 64, 125.
Doubling: 1, 2, 4, 8, 16, 32. Powers of 3: 1, 3, 9, 27, 81, 243.
Second row of differences: if the differences are 3, 5, 9, 15, their own differences are 2, 4, 6, so the next difference is $15 + 8 = 23$ .
Doubling differences mean the same series as $\times 2$ plus a fixed number. Either route gives the same answer, so use whichever you see first.

Worked example

Find the next term: 6, 8, 11, 16, 23, ?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 18 Sep 2025, Shift 3 · Q6Easy

Example 2 · Series · Difference Series

What comes next: 13, 16, 21, 30, 45, ?

Counting 1 or 9 as a prime

1 is not a prime, and neither are 9, 15, 21 or 27. The primes after 7 are 11 and 13, so a difference row of 5, 7, 9 is not the primes.

Skipping a prime

After 31 come 37, 41, 43, 47: the gaps between primes are uneven, so list them, do not add a fixed step.

Drill 19 more on differences that follow their own rule

Concept 3 of 3: Two rules taking turns

If the differences zigzag, such as +8,

-3

, +8,

-3

, the series is two rules taking turns. Often it is easier to see it as two separate series woven together: the odd places and the even places.

Definition

Zigzag differences: look at every other difference. +8, $-3$ , +8, $-3$ means the two steps repeat in turn.
Two series in one: write the 1st, 3rd, 5th terms as one series and the 2nd, 4th, 6th as another. Each has its own simple rule.
Signs that alternate with a growing size: $+1, -3, +5, -7$ goes on as $+9$ .
Count the blank's place to know which of the two series it belongs to.

Worked example

Find the missing term: 12, 20, 17, 25, ?, 30, 27

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 24 Sep 2025, Shift 1 · Q3Easy

Example 3 · Series · Difference Series

What should come in place of the question mark '?' in the following number series? 61, 60, 63, 58, 65, ?

Putting the blank in the wrong series

In a woven series the 7th term belongs to the odd places. Count the places before you decide which rule to continue.

Drill 10 more on two rules taking turns

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 (1)

Steady and evenly growing differences
Next term when the difference grows by a fixed step
$t_{n+1} = t_n + d_n \qquad d_{n+1} = d_n + k$

Watch out for (5)

Not checking the term after the blank
→ Steady and evenly growing differences
Mixing up the sign of the difference
→ Steady and evenly growing differences
Counting 1 or 9 as a prime
→ Differences that follow their own rule
Skipping a prime
→ Differences that follow their own rule
Putting the blank in the wrong series
→ Two rules taking turns

Test yourself on Series

20 past SSC CGL questions from this chapter, timed at 12 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Series chapter test Drill all 75 questions