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Progression

Practice MCQs

Progression topic questions are based on number patterns, sequences, arithmetic progression, geometric progression, harmonic progression, and series.

Quantitative Aptitude Progression Competitive Exams

Progression is an important quantitative aptitude topic based on number patterns, sequences, arithmetic progression, geometric progression, harmonic progression, and series. These concepts are frequently asked in competitive exams, banking exams, aptitude tests, and entrance examinations.

What is Progression?

A progression is an ordered list of numbers that follows a definite rule. Each number in the list is called a term. The terms may increase, decrease, multiply, divide, or follow some fixed relationship.

In aptitude exams, progression questions usually test your ability to identify the pattern, find the next term, calculate the \(n^{th}\) term, or find the sum of a given number of terms.

Quick idea: Arithmetic Progression uses a common difference, while Geometric Progression uses a common ratio.
Type Pattern Example
Arithmetic Progression Difference between consecutive terms is constant 2, 5, 8, 11, 14
Geometric Progression Ratio between consecutive terms is constant 3, 6, 12, 24, 48
Harmonic Progression Reciprocals form an arithmetic progression 1, \( \frac{1}{2} \), \( \frac{1}{3} \), \( \frac{1}{4} \)
Series Sum of terms of a sequence 2 + 4 + 6 + 8

“In progression questions, first identify whether the pattern is based on difference, ratio, or reciprocal.”

Aptitude Tip
Key points
  • Each number in a progression is called a term.
  • AP has a constant difference.
  • GP has a constant ratio.
  • HP is connected with reciprocals.
  • Series means sum of terms.
  • Formula selection depends on the pattern.
sequence series AP GP HP

Visual Understanding

The following diagrams show how arithmetic and geometric progressions grow differently.

Arithmetic Progression
2 5 8 11 14 +3 +3 +3 +3 Same difference between terms
\[ a,\, a+d,\, a+2d,\, a+3d,\, \ldots \]

In AP, the same number is added or subtracted each time.

Geometric Progression
2 4 8 16 32 Same ratio between terms
\[ a,\, ar,\, ar^2,\, ar^3,\, \ldots \]

In GP, each term is multiplied by the same ratio.

Harmonic Progression
1 1/2 1/3 1/4 Reciprocals form AP: 1, 2, 3, 4
\[ \frac{1}{a},\, \frac{1}{a+d},\, \frac{1}{a+2d},\, \ldots \]

In HP, reciprocals of the terms form an arithmetic progression.

Series as Sum of Terms
2 + 4 + 6 + 8 Sum of terms = Series
\[ 2+4+6+8=20 \]

A series is formed by adding the terms of a sequence.

Important Formulas and Rules

AP General Term
\[ a_n=a+(n-1)d \]

\(a\) is first term, \(d\) is common difference.

AP Sum
\[ S_n=\frac{n}{2}\left[2a+(n-1)d\right] \]

Used to find the sum of first \(n\) terms of AP.

AP Sum Using Last Term
\[ S_n=\frac{n}{2}(a+l) \]

\(l\) is the last term of the AP.

Common Difference
\[ d=a_2-a_1 \]

Difference between two consecutive terms in AP.

GP General Term
\[ a_n=ar^{n-1} \]

\(r\) is the common ratio.

GP Sum
\[ S_n=\frac{a(r^n-1)}{r-1} \]

Used when \(r>1\).

GP Sum Alternative
\[ S_n=\frac{a(1-r^n)}{1-r} \]

Useful when \(r<1\).

Common Ratio
\[ r=\frac{a_2}{a_1} \]

Ratio of two consecutive terms in GP.

Infinite GP Sum
\[ S_{\infty}=\frac{a}{1-r} \]

Valid when \(|r|<1\).

HP Rule
\[ \frac{1}{a_1},\frac{1}{a_2},\frac{1}{a_3} \text{ are in AP} \]

Reciprocals of HP terms form AP.

Arithmetic Mean
\[ AM=\frac{a+b}{2} \]

One arithmetic mean between two numbers.

Geometric Mean
\[ GM=\sqrt{ab} \]

One geometric mean between two positive numbers.

Rule: Check the relation between consecutive terms. If difference is fixed, use AP. If ratio is fixed, use GP. If reciprocals form AP, use HP.

Common Types of Questions

Find the Next Term

Identify the difference or ratio and extend the sequence.

  • AP pattern
  • GP pattern
  • Mixed pattern
  • Alternating pattern
Find the \(n^{th}\) Term

Use the general term formula for AP or GP.

  • AP term formula
  • GP term formula
  • Term position
  • Unknown term
Find Sum of Terms

Use the correct series formula.

  • Sum of AP
  • Sum of GP
  • Finite series
  • Infinite GP
Insert Means

Insert numbers between two given terms.

  • Arithmetic mean
  • Geometric mean
  • Harmonic mean
  • Missing terms
Exam approach: Do not apply formulas immediately. First confirm whether the given sequence is AP, GP, HP, or a special pattern.
Method Bank
To Check AP

Find the difference between consecutive terms.

\[ a_2-a_1=a_3-a_2 \]
To Check GP

Find the ratio of consecutive terms.

\[ \frac{a_2}{a_1}=\frac{a_3}{a_2} \]
To Check HP

Take reciprocals and check AP.

\[ a_1,a_2,a_3 \text{ in HP} \Rightarrow \frac{1}{a_1},\frac{1}{a_2},\frac{1}{a_3} \text{ in AP} \]
To Find Missing Term

Use the pattern and solve for the unknown.

\[ \text{Pattern first, formula next} \]

Tip: In number series questions, check differences, second differences, ratios, squares, cubes, and alternating patterns.

Pattern Identification
Difference? Ratio? Reciprocal? AP GP HP
This simple checking order helps you quickly identify the correct progression type.
\[ \text{Constant difference} \Rightarrow \text{AP} \]
\[ \text{Constant ratio} \Rightarrow \text{GP} \]

Step-by-Step Solving Method

Step What to Do Example Check
Step 1 Write down the given terms clearly. 2, 5, 8, 11
Step 2 Check the difference between terms. \(5-2=3,\;8-5=3\)
Step 3 If difference is constant, use AP formulas. Here \(d=3\)
Step 4 If difference is not constant, check ratio. 2, 4, 8, 16 has \(r=2\)
Step 5 Use the correct formula based on the question. Term formula or sum formula
Important: In progression questions, identifying the pattern correctly is more important than memorizing many formulas.

Solved Examples

Question Method Answer
Find the 10th term of the AP: 3, 7, 11, 15, ... Here \(a=3\), \(d=4\), \(n=10\).
\[ a_n=a+(n-1)d \]
\[ a_{10}=3+(10-1)4=3+36=39 \]
39
Find the sum of first 20 terms of the AP: 2, 5, 8, 11, ... Here \(a=2\), \(d=3\), \(n=20\).
\[ S_n=\frac{n}{2}\left[2a+(n-1)d\right] \]
\[ S_{20}=\frac{20}{2}\left[4+19\times3\right] \]
\[ S_{20}=10(4+57)=610 \]
610
Find the 6th term of the GP: 2, 6, 18, 54, ... Here \(a=2\), \(r=3\), \(n=6\).
\[ a_n=ar^{n-1} \]
\[ a_6=2\times3^5=2\times243=486 \]
486
Find the sum of first 5 terms of the GP: 3, 6, 12, 24, ... Here \(a=3\), \(r=2\), \(n=5\).
\[ S_n=\frac{a(r^n-1)}{r-1} \]
\[ S_5=\frac{3(2^5-1)}{2-1}=3(32-1)=93 \]
93
Find the common difference of the AP: 12, 17, 22, 27, ... Difference between consecutive terms:
\[ d=17-12=5 \]
5
Find the common ratio of the GP: 5, 20, 80, 320, ... Ratio between consecutive terms:
\[ r=\frac{20}{5}=4 \]
4
Are 1, \( \frac{1}{3} \), \( \frac{1}{5} \), \( \frac{1}{7} \) in HP? Take reciprocals:
\[ 1,3,5,7 \]
These are in AP with common difference \(2\). 		Yes, they are in HP
Find the arithmetic mean between 8 and 20.
\[ AM=\frac{8+20}{2}=14 \]
14

Note: In exam questions, always identify \(a\), \(d\), \(r\), \(n\), and \(S_n\) carefully before substituting values.

Common Traps and Shortcuts

Common Traps
  • Using AP formula when the sequence is actually GP.
  • Confusing common difference with common ratio.
  • Taking \(n\) as the term value instead of the position.
  • Forgetting that GP formula uses powers.
  • Applying infinite GP sum when \(|r|\) is not less than 1.
  • Not checking reciprocals in HP questions.
Useful Shortcuts
  • AP: subtract consecutive terms.
  • GP: divide consecutive terms.
  • HP: take reciprocals and check AP.
  • For AP, middle term is often average of neighbouring terms.
  • For GP, middle term is often geometric mean of neighbouring terms.
  • For series, first decide whether it is AP or GP.
Exam approach: In progression problems, the fastest method is usually pattern recognition followed by formula substitution.

Practice

A) Multiple Choice Questions
  1. Find the next term: 4, 8, 12, 16, ...
    18 20 22 24
  2. Find the common difference of 7, 11, 15, 19, ...
    2 3 4 5
  3. Find the common ratio of 3, 9, 27, 81, ...
    2 3 6 9
  4. Find the 5th term of AP: 2, 6, 10, 14, ...
    16 18 20 22
  5. The sequence 1, \( \frac{1}{2} \), \( \frac{1}{3} \), \( \frac{1}{4} \) is:
    AP GP HP None
B) Solve the Higher-Order Problems
  1. Find the 15th term of the AP: 5, 9, 13, 17, ... (Hint: Use \(a_n=a+(n-1)d\).)
  2. Find the sum of first 12 terms of the AP: 3, 6, 9, 12, ... (Hint: Use AP sum formula.)
  3. Find the 7th term of the GP: 2, 4, 8, 16, ... (Hint: Use \(a_n=ar^{n-1}\).)
  4. Find the sum of first 6 terms of the GP: 1, 3, 9, 27, ... (Hint: Use GP sum formula.)
  5. Check whether 1, \( \frac{1}{4} \), \( \frac{1}{7} \), \( \frac{1}{10} \) are in HP. (Hint: Take reciprocals.)
C) Match the Concept with the Correct Meaning
Concept Correct Meaning
Arithmetic Progression Common difference is constant
Geometric Progression Common ratio is constant
Harmonic Progression Reciprocals form AP
Series Sum of terms
Arithmetic Mean Average of two numbers
Geometric Mean Square root of product of two numbers
Aptitude Reminder

Progression questions are mainly based on identifying the rule followed by the terms. Check difference for AP, ratio for GP, and reciprocals for HP.

Task: Create five new questions using AP term, AP sum, GP term, GP sum, and HP identification.

Show Suggested Answers
Multiple Choice
  1. 20
    The sequence increases by 4.
    \[ 16+4=20 \]
  2. 4
    \[ d=11-7=4 \]
  3. 3
    \[ r=\frac{9}{3}=3 \]
  4. 18
    Here \(a=2\), \(d=4\), \(n=5\).
    \[ a_5=2+(5-1)4=18 \]
  5. HP
    Reciprocals are:
    \[ 1,2,3,4 \]
    These form an AP.
Higher-Order Problems
  1. AP: \(5,9,13,17,\ldots\). Here \(a=5\), \(d=4\), \(n=15\).
    \[ a_{15}=5+(15-1)4=5+56=61 \]
    Answer = 61.
  2. AP: \(3,6,9,12,\ldots\). Here \(a=3\), \(d=3\), \(n=12\).
    \[ S_{12}=\frac{12}{2}\left[2(3)+(12-1)3\right] \]
    \[ S_{12}=6(6+33)=234 \]
    Answer = 234.
  3. GP: \(2,4,8,16,\ldots\). Here \(a=2\), \(r=2\), \(n=7\).
    \[ a_7=2\times2^6=128 \]
    Answer = 128.
  4. GP: \(1,3,9,27,\ldots\). Here \(a=1\), \(r=3\), \(n=6\).
    \[ S_6=\frac{1(3^6-1)}{3-1} \]
    \[ S_6=\frac{729-1}{2}=364 \]
    Answer = 364.
  5. Given terms: \(1,\frac{1}{4},\frac{1}{7},\frac{1}{10}\). Reciprocals:
    \[ 1,4,7,10 \]
    These are in AP with common difference \(3\).
    Answer = Yes, they are in HP.
Concept Matching
  1. Arithmetic Progression → Common difference is constant
  2. Geometric Progression → Common ratio is constant
  3. Harmonic Progression → Reciprocals form AP
  4. Series → Sum of terms
  5. Arithmetic Mean → Average of two numbers
  6. Geometric Mean → Square root of product of two numbers
Clue Explanation

AP depends on difference, GP depends on ratio, and HP depends on reciprocals. Once the type is identified, choose the correct formula for term or sum.

Exam tips
  • Check common difference first.
  • If difference fails, check common ratio.
  • For HP, take reciprocals and check AP.
  • Use \(a_n\) formula for term-based questions.
  • Use \(S_n\) formula for sum-based questions.
  • Do not use infinite GP formula unless \(|r|<1\).
Practice MCQs