Sequences and Series – General Aptitude for GATE, PSU, ESE, AE/JE | Priyanka Ma’am

Sequences and Series is an important topic in General Aptitude and is frequently asked in GATE, PSU, ESE, AE/JE exams. It focuses on identifying patterns and calculating sums efficiently.

Basic Concepts

A Sequence is an ordered list of numbers, while a Series is the sum of terms of a sequence.

• Sequence: 2, 4, 6, 8, ...
• Series: 2 + 4 + 6 + 8 + ...

Types of Sequences

• Arithmetic Progression (AP)
• Geometric Progression (GP)
• Special Series

Arithmetic Progression (AP)

• nth term: aₙ = a + (n − 1)d
• Sum of n terms: Sₙ = n/2 [2a + (n − 1)d]

Geometric Progression (GP)

• nth term: aₙ = a × rⁿ⁻¹
• Sum of n terms: Sₙ = a(1 − rⁿ)/(1 − r)
• Infinite GP: S = a/(1 − r), where |r| < 1

Important Series

• 1 + 2 + 3 + ... + n = n(n+1)/2
• 1² + 2² + ... + n² = n(n+1)(2n+1)/6
• 1³ + 2³ + ... + n³ = [n(n+1)/2]²

Key Concepts

1. Finding nth Term

Identify pattern (difference or ratio) and apply formula.

2. Missing Term

Check difference, ratio, or alternate patterns.

3. Sum of Series

Use formula instead of adding manually.

Solved Examples

Example 1

Find 10th term of AP: 2, 5, 8, ...

a = 2, d = 3
a₁₀ = 2 + 9×3 = 29

Example 2

Find sum of first 10 natural numbers.

S = 10×11/2 = 55

Example 3

Find 5th term of GP: 2, 4, 8, ...

a = 2, r = 2
a₅ = 2 × 2⁴ = 32

Example 4

Find sum of infinite GP: 3 + 1.5 + 0.75 + ...

a = 3, r = 0.5
S = 3/(1 − 0.5) = 6

Shortcut Tricks

• Identify AP vs GP quickly
• Use standard formulas directly
• Check pattern differences
• Use approximation for faster solving

Questions

Q1. Find sum of first 20 natural numbers.

= 20×21/2 = 210

Q2. Find next term: 2, 6, 18, 54, ?

Multiply by 3 → 162

Exam Tips

• Memorize all formulas
• Practice number series regularly
• Identify pattern quickly
• Avoid manual calculations

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