Progressions Class 11: Complete Guide, All Formulas & Free PDF Download (JEE & Boards)
Progressions — Competishun
Progressions Class 11: Complete Guide, All Formulas & Free PDF Download (JEE & Boards)
Progressions (Sequences and Series) is one of the most fundamental and scoring chapters in Class 11 Maths. It carries significant weightage in JEE and is a guaranteed source of marks in board exams. The chapter deals with ordered lists of numbers that follow a definite pattern, and the sums of these lists.
This chapter covers Arithmetic Progression (AP) (constant difference), Geometric Progression (GP) (constant ratio), Harmonic Progression (HP) (reciprocals in AP), Arithmetic Mean (AM), Geometric Mean (GM), Harmonic Mean (HM), and the relationships between these means. It also covers the sum of finite and infinite series, which is essential for solving advanced problems in JEE.
This page gives you the complete guide to Progressions with all concepts explained in depth. You will find clear definitions, formulas, visual diagrams, and practice questions. Download the free PDF below and keep it handy for quick revision before your JEE Main, board exams, or any other test.
Download the Progressions Complete Guide PDF
Get all Progressions concepts, formulas for AP, GP, HP, AM, GM, HM, infinite series, and practice questions in one clean PDF, free. Perfect for JEE and board revision.
Download Free PDFWhat are Progressions?
The main types of progressions covered in Class 11 are:
Arithmetic Progression (AP)
A sequence where the difference between consecutive terms is constant. Example: 2, 4, 6, 8, ...
Geometric Progression (GP)
A sequence where the ratio between consecutive terms is constant. Example: 2, 4, 8, 16, ...
Harmonic Progression (HP)
A sequence where the reciprocals of the terms are in AP. Example: 1, 1/2, 1/3, 1/4, ...
Glossary of Progressions Terms — Complete A to Z
Before diving deep into each topic, let's understand the key terminology used in this chapter:
| Term | Definition |
|---|---|
| Sequence | An ordered list of numbers. Example: a₁, a₂, a₃, ... |
| Term | Each number in a sequence. Denoted by aₙ (nth term). |
| Series | The sum of the terms of a sequence. Denoted by Sₙ. |
| Arithmetic Progression (AP) | A sequence where the difference between consecutive terms is constant (d). |
| Common Difference (d) | The constant difference in an AP: d = aₙ - aₙ₋₁. |
| Geometric Progression (GP) | A sequence where the ratio between consecutive terms is constant (r). |
| Common Ratio (r) | The constant ratio in a GP: r = aₙ / aₙ₋₁. |
| Harmonic Progression (HP) | A sequence where the reciprocals of the terms are in AP. |
| Arithmetic Mean (AM) | The average of two or more numbers: AM = (a + b) / 2. |
| Geometric Mean (GM) | The nth root of the product of n numbers: GM = √(ab). |
| Harmonic Mean (HM) | The reciprocal of the arithmetic mean of the reciprocals: HM = 2ab/(a+b). |
| Infinite Series | A series with an infinite number of terms. Its sum exists only under certain conditions. |
| Mastering these terms is essential for understanding Progressions. They will be used throughout this guide. | |
Arithmetic Progression (AP) — Constant Difference
Key Formulas
| Quantity | Formula | Description |
|---|---|---|
| nth Term | aₙ = a + (n-1)d | Find the nth term given the first term (a) and common difference (d).[reference:0] |
| Sum of n Terms | Sₙ = n/2 [2a + (n-1)d] | Sum of the first n terms.[reference:1][reference:2] |
| Sum of n Terms (Alternate) | Sₙ = n/2 (a + l) | Where l is the last term (nth term). |
| Common Difference | d = aₙ - aₙ₋₁ | The constant difference between consecutive terms. |
| nth Term from End | aₙ = l - (n-1)d | Where l is the last term. |
| The formula for the nth term is the most frequently used in AP problems. Memorise it and practise applying it.[reference:3] | ||
Properties of AP
- Three terms in AP: If a, b, c are in AP, then 2b = a + c.[reference:4]
- nth term of AP: aₙ = a + (n-1)d
- Sum of n terms: Sₙ = n/2 [2a + (n-1)d]
- Sum of first and last terms: a₁ + aₙ = a₂ + aₙ₋₁ = constant
Geometric Progression (GP) — Constant Ratio
Key Formulas
| Quantity | Formula | Description |
|---|---|---|
| nth Term | aₙ = arⁿ⁻¹ | Find the nth term given the first term (a) and common ratio (r).[reference:7] |
| Sum of n Terms (r ≠ 1) | Sₙ = a(rⁿ - 1) / (r - 1) | Sum of the first n terms.[reference:8] |
| Sum of n Terms (r < 1) | Sₙ = a(1 - rⁿ) / (1 - r) | Alternate form when |r| < 1. |
| Sum of Infinite GP (|r| < 1) | S∞ = a / (1 - r) | Sum of an infinite geometric series.[reference:9] |
| Common Ratio | r = aₙ / aₙ₋₁ | The constant ratio between consecutive terms.[reference:10] |
| The sum of an infinite GP exists only when |r| < 1.[reference:11] | ||
Properties of GP
- Three terms in GP: If a, b, c are in GP, then b² = ac.[reference:12]
- nth term of GP: aₙ = arⁿ⁻¹
- Sum of n terms: Sₙ = a(rⁿ - 1) / (r - 1), r ≠ 1
- Product of first and last terms: a₁ × aₙ = a₂ × aₙ₋₁ = constant
Harmonic Progression (HP) — Reciprocals in AP
Key Formulas
| Quantity | Formula | Description |
|---|---|---|
| nth Term of HP | aₙ = 1 / [1/a + (n-1)d] | Where a is the first term and d is the common difference of the corresponding AP. |
| Three terms in HP | 2/b = 1/a + 1/c | If a, b, c are in HP, then the reciprocals are in AP. |
| There is no direct formula for the sum of an HP. Problems are usually solved by converting the HP to an AP. | ||
Arithmetic Mean (AM), Geometric Mean (GM), Harmonic Mean (HM)
Relationship Between AM, GM, and HM
For two positive numbers, the following relationship holds:[reference:18][reference:19]
AM ≥ GM ≥ HM (for positive numbers)[reference:20]
Insertion of Means — Between Two Numbers
Inserting Means in AP
To insert n arithmetic means between a and b, the common difference is:
The means are: a + d, a + 2d, ..., a + nd
Inserting Means in GP
To insert n geometric means between a and b, the common ratio is:
The means are: ar, ar², ..., arⁿ
Inserting Means in HP
To insert n harmonic means between a and b, first convert to AP, insert means, then convert back.
Special Series — Sum of Squares and Cubes
| Series | Sum Formula |
|---|---|
| Sum of first n natural numbers | Σn = n(n+1)/2 |
| Sum of squares of first n natural numbers | Σn² = n(n+1)(2n+1)/6 |
| Sum of cubes of first n natural numbers | Σn³ = [n(n+1)/2]² |
| These formulas are essential for solving problems involving sums of series. | |
Practice Questions — From JEE and Boards
| Question | Answer |
|---|---|
| Q1: Find the 10th term of the AP: 2, 5, 8, 11, ... | a = 2, d = 3. a₁₀ = 2 + (9)(3) = 29. |
| Q2: Find the sum of the first 20 terms of the AP: 1, 4, 7, 10, ... | a = 1, d = 3, n = 20. S₂₀ = 20/2 [2(1) + (19)(3)] = 10(2 + 57) = 590. |
| Q3: Find the 6th term of the GP: 3, 6, 12, 24, ... | a = 3, r = 2. a₆ = 3 × 2⁵ = 3 × 32 = 96. |
| Q4: Find the sum of the infinite GP: 1 + 1/2 + 1/4 + 1/8 + ... | a = 1, r = 1/2. S∞ = 1 / (1 - 1/2) = 1 / (1/2) = 2.[reference:21] |
| Q5: If a, b, c are in AP, what is the value of 2b? | 2b = a + c.[reference:22] |
| Q6: If a, b, c are in GP, what is the value of b²? | b² = ac.[reference:23] |
| Q7: Find the AM and GM of 4 and 16. | AM = (4+16)/2 = 10. GM = √(4×16) = √64 = 8. |
| Q8: Find the sum of the series: 1² + 2² + 3² + ... + 10². | Σn² = 10(11)(21)/6 = 385. |
| Practise these types of questions to become comfortable with applying Progressions concepts in exam scenarios. | |
All Progressions Formulas at a Glance
Arithmetic Progression (AP)
aₙ = a + (n-1)d
Sₙ = n/2 [2a + (n-1)d]
Sₙ = n/2 (a + l)
Geometric Progression (GP)
aₙ = arⁿ⁻¹
Sₙ = a(rⁿ - 1)/(r - 1)
S∞ = a/(1 - r) (|r| < 1)
Harmonic Progression (HP)
Reciprocals form an AP
aₙ = 1/[1/a + (n-1)d]
Means
AM = (a+b)/2
GM = √(ab)
HM = 2ab/(a+b)
AM ≥ GM ≥ HM
AM × HM = GM²
Special Series
Σn = n(n+1)/2
Σn² = n(n+1)(2n+1)/6
Σn³ = [n(n+1)/2]²
Common Mistakes in Progressions
- Confusing AP and GP: AP has a constant difference (d), while GP has a constant ratio (r).
- Forgetting the condition for infinite GP: The sum of an infinite GP exists only when |r| < 1.[reference:24]
- Using the wrong formula for sum of GP: Sₙ = a(rⁿ - 1)/(r - 1) is for r > 1; Sₙ = a(1 - rⁿ)/(1 - r) is for r < 1.
- Misapplying the AM-GM inequality: AM ≥ GM ≥ HM is only for positive numbers.[reference:25]
- Forgetting to convert HP to AP: Always convert HP to AP by taking reciprocals before solving.
- Not checking the common difference/ratio: Always verify that the difference or ratio is constant before identifying a sequence as AP or GP.
Why Progressions Matters for JEE and Boards
- Foundation for higher mathematics: Progressions are the building block for calculus, algebra, and many other topics.
- High weightage: This chapter appears in 1-2 questions in JEE Main and is a guaranteed source of marks in board exams.
- Conceptual clarity: This chapter rewards students who understand the definitions and properties rather than just memorizing formulas.
- Practical relevance: Progressions are used everywhere, from finance (compound interest) to physics (growth and decay).
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Frequently Asked Questions — Progressions
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