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Series and Progressions
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Series and Progressions
Series and Progressions are important topics in the Quantitative Aptitude section of most placement tests and competitive exams. These topics test a candidate’s ability to recognise patterns, apply formulas, and solve numerical sequences efficiently.
Candidates are usually required to:
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Identify the pattern in a series
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Find the next or missing term
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Calculate the sum of terms
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Determine the number of terms
The main types of progressions are:
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Arithmetic Progression (AP)
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Geometric Progression (GP)
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Harmonic Progression (HP)
Arithmetic Progression (AP)
Definition
An Arithmetic Progression is a sequence of numbers in which the difference between consecutive terms is constant. This constant value is called the common difference (d).
Important Formulas
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nth term:
aₙ = a + (n − 1)d -
Sum of first n terms:
Sₙ = (n/2)[2a + (n − 1)d]
Where:
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a = first term
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d = common difference
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n = number of terms
Example 1
Find the sum of the first 50 terms of an AP where:
a = 5, d = 3
Sₙ = (50/2)[2(5) + (49)(3)]
= 25[10 + 147]
= 25 × 157
= 3925
Example 2
Find the 15th term of an AP where:
a = 7, d = −2
a₁₅ = 7 + (15 − 1)(−2)
= 7 − 28
= −21
Geometric Progression (GP)
Definition
A Geometric Progression is a sequence in which the ratio between consecutive terms is constant. This constant value is called the common ratio (r).
Important Formulas
-
nth term:
aₙ = a × r⁽ⁿ⁻¹⁾ -
Sum of first n terms:
If r ≠ 1:
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Sₙ = a(1 − rⁿ) / (1 − r) (for r < 1)
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Sₙ = a(rⁿ − 1) / (r − 1) (for r > 1)
Example 1
Find the sum of the first 6 terms of a GP where:
a = 3, r = 2
S₆ = 3(2⁶ − 1)/(2 − 1)
= 3(64 − 1)
= 3 × 63
= 189
Example 2
Find the 10th term of a GP where:
a = 2, r = 0.5
a₁₀ = 2 × (0.5)⁹
= 2 × (1/512)
= 1/256
Harmonic Progression (HP)
Definition
A Harmonic Progression is a sequence in which the reciprocals of the terms form an Arithmetic Progression.
If the reciprocals follow an AP, the original sequence is called an HP.
General Term
If the reciprocals form an AP:
aₙ = 1 / [a + (n − 1)d]
Where:
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a = first term of the AP formed by reciprocals
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d = common difference
Example 1
Find the 8th term of an HP where:
a = 1, d = 1/2
a₈ = 1 / [1 + (8 − 1)(1/2)]
= 1 / (1 + 7/2)
= 1 / (9/2)
= 2/9
Example 2
If the reciprocals of an HP form the AP:
1, 2, 3, 4, 5
Then the HP is:
1, 1/2, 1/3, 1/4, 1/5
Tips for Solving Series and Progression Problems
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First identify the type of progression: AP, GP, or HP.
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Check whether the difference or ratio is constant.
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Write down the known terms.
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Find the common difference or ratio.
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Use the appropriate formula.
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Practise different types of questions to improve speed and accuracy.
