SSC CGL Reasoning · Series

Multiplication Series

When the terms grow or shrink fast, divide each term by the one before it: a steady ratio, a ratio with a small amount added, a multiplier that rises by one, or two operations taking turns.

All Series notes

Why this matters

Fifty-two questions in the bank, 34 of them from the 2025 shifts: 37 EASY, 14 MODERATE and the one HARD question of the chapter. Ten have a steady ratio, fifteen multiply by a fixed number and then add or take away a fixed amount, seventeen change the multiplier or the amount added at each step, and ten mix operations or build each term from earlier ones.

Concept 1 of 4: A steady ratio

If every term is the one before times the same number, the series is a geometric progression. Divide one pair, check another pair, and keep multiplying.

Definition

When to divide: if the terms more than double, or fall by half or more, differences grow too fast to help. Divide instead.
Steady ratio: $\frac{21}{7} = \frac{63}{21} = 3$ , so each term is 3 times the one before.
Ratios below 1 work the same way: 625, 125, 25 divides by 5 each time; 120, 96 is times 0.8.
Term n directly: the first term times the ratio raised to $n - 1$ .

Term n of a geometric progression

T_n = a\,r^{\,n-1}

$a$ the first term
$r$ the common ratio
$n$ the place of the term

Worked example

Find the next term: 7, 21, 63, 189, ?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 13 Sep 2025, Shift 2 · Q8Easy

Example 1 · Series · Multiplication Series

Complete the pattern: 120, 96, 76.8, 61.44, ?

Raising the ratio to the power n

The 3rd term is

a r^2

, not

a r^3

: the first term has had no multiplication yet.

Drill 9 more on a steady ratio

Concept 2 of 4: Multiply, then add or take away a fixed amount

Many series are almost geometric: each term is about double, or about triple, the last. The small amount left over is the same every time, so the rule is times a, plus b.

Definition

Rough ratio first: 3, 8, 23, 68 is roughly times 3 each time.
Find the leftover: $3 \times 3 = 9$ , and 8 is one less, so the rule is $\times 3 - 1$ .
Check it on a second pair before trusting it: $8 \times 3 - 1 = 23$ .
Halving works the same way: $\div 2 + 2$ turns 100 into 52.
Quick check for times 2: in a $\times 2 + b$ series the differences double (and for times 3 they triple), so the difference route gives the same answer.

Times a fixed number, plus a fixed amount

t_{n+1} = a\,t_n + b

$a$ the fixed multiplier (about the ratio of next-door terms)
$b$ the fixed amount added (negative if taken away)

Worked example

Find the next term: 3, 8, 23, 68, ?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 16 Sep 2025, Shift 2 · Q8Easy

Example 2 · Series · Multiplication Series

What will come at the place of question mark? 4, 10, 22, 46, 94, ?

Checking the rule on one pair only

From 5 to 11 alone, both

\times 2 + 1

and

\times 3 - 4

work. Test the rule on a second pair; only one survives.

Drill 14 more on multiply, then add or take away a fixed amount

Concept 3 of 4: The multiplier or the amount added changes

Here the rough ratios are not steady: they go 2, 3, 4, 5. The series multiplies by a number that rises by one, and sometimes adds a number that rises too.

Definition

List the rough ratios: for 7, 22, 89 they are about 3 and about 4, so the multipliers rise by one.
Common forms: $\times 1, \times 2, \times 3$ and so on; $\times n + n$ ( $\times 1 + 1$ , $\times 2 + 2$ ); $\times 2$ plus 1, 2, 3; $\times 2 - 1$ , $\times 3 - 2$ , $\times 4 - 3$ .
Shrinking series: 720, 720, 360, 120 divides by 1, 2, 3, so the next divisor is 4.
Work out each leftover (term minus multiplier times the term before) and look at the row of leftovers; it is steady, rising by one, or taking turns.

A multiplier that rises with each step

t_{n+1} = (m + n)\,t_n + c_n

$m + n$ the multiplier, one more at each step
$c_n$ the amount added at that step (steady, rising or taking turns)

Worked example

Find the next term: 2, 3, 8, 27, 112, ?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 17 Sep 2025, Shift 2 · Q7Moderate

Example 3 · Series · Multiplication Series

Find the missing term: 7, 22, 89, ________, 2677, 18740

Forcing a fixed multiplier

In 1, 2, 6, 24 the ratios are 2, 3, 4. Trying a steady

\times 3

plus something leaves a messy row of leftovers. When the rough ratios climb, the multiplier climbs.

Using the wrong multiplier for the missing step

When the blank is in the middle, its step uses the multiplier for that place. Count the steps from the start so you do not use the one meant for the last pair.

Drill 16 more on the multiplier or the amount added changes

Concept 4 of 4: Operations that take turns, and terms built from earlier terms

Two kinds of series break the patterns above: those that switch between two operations, such as times 2 then add 3, and those where each term is made from the terms before it.

Definition

Operations that take turns: check the steps one at a time. 4, 12, 15, 45, 48 is $\times 3$ , $+3$ , $\times 3$ , $+3$ .
Sum of the two before (Fibonacci type): 2, 5, 7, 12, 19, each term is the two before added.
Product of the two before: 2, 3, 6, 18, each term is the two before multiplied.
Square of the term before, plus 1: 3, 10, 101.
Signal: differences that jump about with no pattern, or ratios that are whole numbers only at every other step.

Worked example

Find the next term: 4, 12, 15, 45, 48, ?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 26 Sep 2025, Shift 3 · Q21Easy

Example 4 · Series · Multiplication Series

Complete the series. 7, 14, 12, 24, 22, ?

Using differences on a series that takes turns

For 4, 12, 15, 45 the differences 8, 3, 30 look random. Look at each step's operation instead: a whole-number ratio at every other step is the giveaway.

Drill 9 more on operations that take turns, and terms built from earlier terms

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 (3)

A steady ratio
Term n of a geometric progression
$T_n = a\,r^{\,n-1}$
Multiply, then add or take away a fixed amount
Times a fixed number, plus a fixed amount
$t_{n+1} = a\,t_n + b$
The multiplier or the amount added changes
A multiplier that rises with each step
$t_{n+1} = (m + n)\,t_n + c_n$

Watch out for (5)

Raising the ratio to the power n
→ A steady ratio
Checking the rule on one pair only
→ Multiply, then add or take away a fixed amount
Forcing a fixed multiplier
→ The multiplier or the amount added changes
Using the wrong multiplier for the missing step
→ The multiplier or the amount added changes
Using differences on a series that takes turns
→ Operations that take turns, and terms built from earlier terms

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