Progressions Explained with Arithmetic and Geometric Series
Progression is a sequence of numbers that follows a fixed mathematical pattern. The main types are Arithmetic Progression (AP), Geometric Progression (GP) and Harmonic Progression (HP). This page covers their terms, sums, means, properties, progression formulas and practical methods for solving progression aptitude questions.
What is Progression?
A progression is written as a, a₂, a₃, ..., aₙ, where aₙ denotes the nth term. In an AP, the difference between consecutive terms is fixed. In a GP, the ratio of consecutive terms is fixed. In an HP, the reciprocals of the terms form an AP. For example, 5, 8, 11, 14 is an AP with common difference 3, while 2, 6, 18, 54 is a GP with common ratio 3.
Progression Video Lessons
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Special Sums: n, n² and n³ Formulas
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Progression Formula & Tricks
Important Formulas
Here, a is the first term, d is the common difference and n is the term number.
Here, l is the last term. Use the second form when the first and last terms are known.
Here, a is the first term and r is the common ratio.
The equivalent form Sₙ = a(1 − rⁿ)/(1 − r) can be used when it gives simpler signs.
An infinite GP has a finite sum only when the absolute value of its common ratio is less than 1.
The reciprocal sequence has a constant difference.
For positive x and y, AM, GM and HM are the arithmetic, geometric and harmonic means respectively.
Quick Tricks
The sum of an AP can be found by multiplying the number of terms by the average of the first and last terms.
For an AP with an odd number of terms, the middle term is the average of the first and last terms.
For three consecutive positive terms of a GP, the middle term is the geometric mean of the other two terms.
For a finite GP, use the standard sum formula rather than adding every term. For r = 2, the sum is a(2ⁿ − 1).
Progression Concepts
Arithmetic Progression
The constant difference is called the common difference and is denoted by d. If the first term is a, the nth term is aₙ = a + (n − 1)d. The sum of n terms is Sₙ = n/2[2a + (n − 1)d]. The common difference may be positive, negative or zero.
Geometric Progression
The constant ratio is denoted by r. For first term a, the nth term is aₙ = arⁿ⁻¹. The sum of n terms is Sₙ = a(rⁿ − 1)/(r − 1), where r ≠ 1. If |r| < 1, the infinite sum is S∞ = a/(1 − r).
Harmonic Progression
If x₁, x₂, x₃, ... is an HP, then 1/x₁, 1/x₂, 1/x₃, ... has a constant difference. If the reciprocal AP has first term p and common difference d, the nth HP term is 1/[p + (n − 1)d]. For positive numbers x and y, their harmonic mean is 2xy/(x + y).
Arithmetic, Geometric and Harmonic Means
These means satisfy AM ≥ GM ≥ HM, with equality only when x = y. If A, G and H are the three means of x and y, then G² = AH. An arithmetic mean gives an AP of three terms, a geometric mean gives a GP of three terms, and a harmonic mean gives an HP of three terms.
Finding Missing Terms in a Progression
For an AP, subtract consecutive known terms to find d. For a GP, divide consecutive known terms to find r. For an HP, first take reciprocals and then apply the AP method. If two terms of an AP are known, use aₙ − aₘ = (n − m)d; if two terms of a GP are known, divide their nth-term equations to obtain rⁿ⁻ᵐ.
Progression Data Table
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Comparison of Common Progressions
Definitions, nth-term formulas and sum formulas for AP, GP and HP.
| Type | Defining rule | Nth term | Finite sum |
|---|---|---|---|
| AP | Constant difference d | a + (n − 1)d | n/2[2a + (n − 1)d] |
| GP | Constant ratio r | arⁿ⁻¹ | a(rⁿ − 1)/(r − 1), r ≠ 1 |
| HP | Reciprocals form an AP | 1/[p + (n − 1)d] | Find by summing the reciprocal-form terms individually |
Note: In the HP row, p and d are the first term and common difference of the reciprocal AP, not of the HP itself.
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Progression Revision Points
Key progression formulas and properties for quick revision.
- AP has a constant difference: aₙ = a + (n − 1)d.
- The AP sum is Sₙ = n/2[2a + (n − 1)d] = n(a + l)/2.
- GP has a constant ratio: aₙ = arⁿ⁻¹.
- The finite GP sum is Sₙ = a(rⁿ − 1)/(r − 1), for r ≠ 1.
- The infinite GP sum exists only when |r| < 1 and equals a/(1 − r).
- HP is identified by taking reciprocals and checking for an AP.
- For positive x and y, AM ≥ GM ≥ HM and G² = AH.
- The middle term of an odd-term AP is the average of the end terms.
Progression FAQs
What is the nth term of an Arithmetic Progression?
The nth term is aₙ = a + (n − 1)d, where a is the first term and d is the common difference.
How is the sum of an AP calculated when the first and last terms are known?
Use Sₙ = n(a + l)/2, where n is the number of terms, a is the first term and l is the last term. For example, the sum of 2, 5, 8, 11 is 4(2 + 11)/2 = 26.
What is the difference between an AP and a GP?
An AP has a constant difference between consecutive terms, whereas a GP has a constant ratio. For example, 3, 7, 11 is an AP and 3, 6, 12 is a GP.
When can the sum of an infinite GP be calculated?
It has a finite sum only when |r| < 1. The sum is S∞ = a/(1 − r); for 8 + 4 + 2 + ..., it is 8/(1 − 1/2) = 16.
How can an HP be identified?
Take the reciprocal of every term. If the reciprocals form an AP, the original sequence is an HP. For example, 1/3, 1/6, 1/9 is an HP because 3, 6, 9 is an AP.
What is the relation among AM, GM and HM?
For positive numbers, AM ≥ GM ≥ HM. Also, if A, G and H are their respective means, then G² = AH.