Sequences & Progressions

medium importance~1 Q in Tier 123 formulas⚡ 15 shortcuts5 subtopics

Overview Learn (5)Formulas (23)Examples (5)Practice (65)

Progressions are lists that follow a fixed rule: APs add a constant, GPs multiply by one, HPs hide an AP in their reciprocals, and special series come with ready-made sum formulas. Alongside term-pattern sequences, this block gives a quick mark in SSC, Railway and Banking papers and anchors the series questions of CAT algebra.

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Learn — 5 subtopics

One page per subtopic: detailed notes, every question type, formulas, tricks and practice sets.

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Formula sheet

Every formula on one printable page, grouped by subtopic.

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Solved examples

5 exam-level questions worked step by step.

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Practice & test

65 questions — untimed practice or a timed test with analysis.

Track record in the exam

Test difficulty mix (65 questions)

21 easy31 medium13 hard

Question patterns exams keep repeating

Taken from previous-year papers. If a pattern is marked "very common", expect to see it in your exam.

nth term or sum of an AP

very common

Rows of seats, monthly savings, wage steps — a quantity changes by the same amount each step, and a term or a total is asked.

Name $a$ , $d$ and $n$ from the story.

Use $a + (n-1)d$ for one term, $\frac{n}{2}[2a + (n-1)d]$ for the total.

Example

A hall has 18 rows of chairs; the first row has 15 chairs and each row has 3 more than the previous row. How many chairs in all?

$S_{18} = \frac{18}{2}[2 \times 15 + 17 \times 3] = 9 \times 81$ = 729

Learn this in “Arithmetic Progressions” →

Middle term of an AP

common

The sum of an odd number of AP terms is given, and the middle term — or a term named by its middle position — is asked.

Middle term = sum ÷ count.

For three terms, name them a − d, a, a + d.

Example

The sum of 15 terms of an AP is 375. Find the 8th term.

Middle $= \frac{375}{15}$ = 25

Learn this in “Arithmetic Progressions” →

Inserting arithmetic means

occasional

'Insert k arithmetic means between a and b' — the means, their gap, or one of them is asked.

Gap = (b − a)/(k + 1).

Build the means by adding the gap repeatedly.

Example

Insert 4 arithmetic means between 8 and 23. Find the largest mean.

$d = \frac{15}{5} = 3$ ; means 11, 14, 17, 20 → largest is 20

Learn this in “Arithmetic Progressions” →

GP term or finite sum

very common

A quantity multiplies by the same factor each step (doubling, tripling) and a later term or a total is asked.

One term: $ar^{n-1}$ .

Total: $\frac{a(r^n - 1)}{r - 1}$ ; if $r = 1$ , the total is $na$ .

Example

Find the sum of the first 6 terms of 4, 12, 36, ...

$\frac{4(3^6 - 1)}{2} = 2 \times 728$ = 1456

Learn this in “Geometric Progressions” →

Sum to infinity

common

An endless list with ratio between −1 and 1 (8 + 4 + 2 + ...) and its total is asked.

Check $|r| < 1$ , then use $\frac{a}{1-r}$ .

Reverse use: given the sum, find $a$ or $r$ .

Example

Find the sum 8 + 4 + 2 + 1 + ...

$\frac{8}{1 - 1/2}$ = 16

Learn this in “Geometric Progressions” →

HP through reciprocals

common

The list has unit fractions like 1/2, 1/5, 1/8 — and a term or term-number is asked.

Flip the list into its AP.

Solve in the AP, flip the final answer back.

Example

Which term of the HP 1/2, 1/5, 1/8, ... is 1/32?

AP of reciprocals: 2, 5, 8, ...; $2 + (n-1)3 = 32$ , so $n$ = 11

Learn this in “Harmonic Progressions & AGP” →

AGP series sums

occasional

Coefficients grow by addition while powers grow by multiplication: 1 + 2·2 + 3·4 + 4·8.

Multiply S by r, line up the powers, subtract.

Infinite case: $\frac{a}{1-r} + \frac{dr}{(1-r)^2}$ .

Example

Find the sum 1 + 2×2 + 3×4 + 4×8 + 5×16.

$S - 2S = 1 + 2 + 4 + 8 + 16 - 160 = -129$ , so $S$ = 129

Learn this in “Harmonic Progressions & AGP” →

Direct power sums (Σn, Σn², Σn³)

very common

A run of natural numbers, squares or cubes is to be added, often starting past 1.

Apply the matching power-sum formula.

Runs that start past 1: subtract two power sums.

Example

Find the sum 21 + 22 + ... + 50.

$\frac{50 \times 51}{2} - \frac{20 \times 21}{2} = 1275 - 210$ = 1065

Learn this in “Special Series & Standard Sums” →

Telescoping fraction sums

common

A chain of fractions like 1/(1·2) + 1/(2·3) + ... — the denominators overlap.

Split each term as a difference of two simple fractions.

Cancel; only the ends survive.

Example

Find the sum 1/(1·2) + 1/(2·3) + ... + 1/(9·10).

Split: $\frac{1}{k(k+1)} = \frac{1}{k} - \frac{1}{k+1}$ ; sum $= 1 - \frac{1}{10}$ = 9/10

Learn this in “Special Series & Standard Sums” →

Next, missing or wrong term in a series

very common

A bare list of numbers with a '?' or 'find the wrong number' — the reasoning staple of SSC, Railway and Banking papers.

Run the routine: differences, second differences, ratios, alternating chains, position rules.

A wrong number must break the rule that fits every other term.

Example

Find the wrong number: 2, 6, 12, 20, 30, 40, 56.

Rule $n(n+1)$ gives 2, 6, 12, 20, 30, 42, 56 — so 40 is wrong

Learn this in “Pattern Sequences: Next, Missing & Wrong Terms” →

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