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AFCAT Sequences & Series

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This page covers AFCAT Sequences & Series with complete concept notes, 39 graded practice MCQs, key points and exam-specific tips. Free to study.

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Concept Notes

Sequences & Series— Rules & Concept

1.Core Concept

Read this first — the foundation of the topic
Core Types

Arithmetic Progression (AP) where difference between consecutive terms is constant. Geometric Progression (GP) where ratio between consecutive terms is constant. Harmonic Progression (HP) where reciprocals form an AP

Arithmetic Progression Formulas

- nth term: an = a + (n-1)d where a is first term, d is common difference - Sum of n terms: Sn = n/2[2a + (n-1)d] or Sn = n/2[first term + last term] - If three terms are in AP: 2b = a + c (middle term is arithmetic mean) Geometric Progression Formulas: - nth term: an = a × r^(n-1) where r is common ratio - Sum of n terms: Sn = a(r^n - 1)/(r - 1) when r ≠ 1 - Sum to infinity: S∞ = a/(1-r) when |r| < 1 - If three terms are in GP: b² = ac (middle term squared equals product of other two)

2.Exam Patterns

What examiners ask — read before attempting PYQs

NDA typically asks for finding nth terms, sum formulas, inserting means between terms, and identifying progression types. Word problems involving time-distance, population growth, and salary increments are common. Powerful Shortcut - The 'Middle Term Trick': In any AP with odd number of terms, the middle term equals the arithmetic mean of all terms. This saves massive calculation time in sum problems.

3.Worked Example

Solve this step-by-step before moving on
1
Step 1

Identify first term a = 3, common difference d = 7-3 = 4

2
Step 2

Apply formula an = a + (n-1)d

3
Step 3

a15 = 3 + (15-1)×4 = 3 + 14×4 = 3 + 56 = 59 Answer: 59 Worked Example 2: Find sum of first 8 terms of GP: 2, 6, 18, 54...

1
Step 1

First term a = 2, common ratio r = 6/2 = 3

2
Step 2

Apply formula Sn = a(r^n - 1)/(r - 1)

3
Step 3

S8 = 2(3^8 - 1)/(3 - 1) = 2(6561 - 1)/2 = 6560 Answer: 6560 Critical Shortcut for GP Sum: When r > 1, use Sn = a(r^n - 1)/(r - 1). When r < 1, use Sn = a(1 - r^n)/(1 - r). This prevents negative number confusion. Another

4.Shortcuts

Use these to save 30–60 seconds per question
#1 MOST COMMON TRAP

Students confuse GP sum formulas. They use Sn = a(r^n - 1)/(r - 1) when r < 1, leading to calculation errors. Always check if r > 1 or r < 1 first. Also, many students forget that HP has no direct formula - you must convert reciprocals to AP first

Harmonic Mean Connection

If a, b, c are in HP, then 1/a, 1/b, 1/c are in AP. Use this conversion technique for all HP problems

Exam Success Strategy

Master the basic formulas first. Then practice identifying progression types quickly. Finally, memorize the shortcuts above - they can save 2-3 minutes per question, giving you crucial extra time for other topics.

Key Points to Remember

  • AP nth term formula: an = a + (n-1)d, where a is first term and d is common difference
  • AP sum formula: Sn = n/2[2a + (n-1)d] or Sn = n/2[first + last term]
  • GP nth term formula: an = a × r^(n-1), where r is common ratio
  • GP sum formula: Sn = a(r^n - 1)/(r - 1) when r ≠ 1
  • Three terms in AP: 2b = a + c (middle term is arithmetic mean)
  • Three terms in GP: b² = ac (middle term squared equals product of extremes)
  • GP infinite sum: S∞ = a/(1-r) when |r| < 1
  • HP conversion trick: If terms are in HP, their reciprocals are in AP
  • Quick AP sum: When you know first and last terms, use Sn = n/2(first + last)
  • GP sum shortcut: Check if r > 1 or r < 1 before applying formula to avoid sign errors

Exam-Specific Tips

  • Sum to infinity of GP exists only when |r| < 1, where r is common ratio
  • Arithmetic mean of a and b is (a+b)/2, geometric mean is √(ab), harmonic mean is 2ab/(a+b)
  • In AP, sum of terms equidistant from beginning and end is constant
  • For GP with first term a and ratio r, sum of first n terms is a(r^n-1)/(r-1)
  • Three numbers a, b, c are in GP if and only if b² = ac
  • Common difference of AP can be found using d = (last term - first term)/(n-1)
  • Geometric mean of positive numbers is always less than or equal to arithmetic mean
  • Sum of first n natural numbers is n(n+1)/2, which forms an AP with a=1, d=1
Practice MCQs

Sequences & Series — Practice Questions

39graded MCQs · easy to hard · full solution & trap analysis · showing 20 of 39

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Practice 1easy

The sum of an infinite geometric series with first term a = 6 and common ratio r = 1/2 is:

Practice 2easy

The nth term of a sequence is given by aₙ = 2n² - 3n + 1. What is the sum of the first 5 terms?

Practice 3easy

In a geometric progression, the first term is 2 and the common ratio is 1/2. What is the sum of the first 6 terms?

Practice 4easy

In an arithmetic progression, the 7th term is 28 and the 15th term is 60. What is the common difference?

Practice 5easy

The sum of an infinite geometric series with first term a = 5 and common ratio r = 1/3 is:

Practice 6easy

The sum of the series 2 + 4 + 8 + 16 + ... up to 10 terms is:

Practice 7easy

If the sum of the first n natural numbers is 55, then n equals:

Practice 8easy

The 3rd term of an arithmetic progression is 7 and the 7th term is 19. What is the sum of the first 10 terms?

Practice 9easy

The sum of an infinite geometric series with first term a and common ratio r is 8. If a = 2, what is the value of r?

Practice 10easy

In an arithmetic progression, the sum of the first n terms is Sₙ = 2n² + 3n. What is the common difference of the progression?

Practice 11easy

The sum of the first n terms of a sequence is given by Sₙ = 3n² + 2n. What is the 5th term of the sequence?

Practice 12easy

In a geometric progression, the first term is 2 and the common ratio is 3. What is the sum of the first 4 terms?

Practice 13easy

An arithmetic progression has first term a = 5 and common difference d = 3. If the sum of the first n terms is 440, find n.

Practice 14medium

The sum of the first n terms of a sequence is given by Sₙ = 3n² + 2n. Find the 5th term of the sequence.

Practice 15medium

In a geometric progression, the 3rd term is 4 and the 6th term is 32. If all terms are positive, what is the sum of the first 5 terms?

Practice 16medium

The sum of an infinite geometric series with first term a and common ratio r is 6. If the first term is doubled and the common ratio is halved, the new sum is 8. Find the original common ratio r.

Practice 17medium

The sum of the first n terms of a sequence is Sₙ = n(n + 1)(n + 2)/3. What is the sum of the 4th and 5th terms?

Practice 18medium

In an arithmetic progression, the sum of the first 10 terms is 100, and the sum of the first 20 terms is 300. Find the 15th term of the progression.

Practice 19medium

In a geometric progression, the 3rd term is 4 and the 6th term is 32. What is the sum of the first 5 terms of this GP?

Practice 20medium

The sum of an infinite geometric series with first term a and common ratio r is 6. If the first term is 2, what is the common ratio?

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60-Second Revision — Sequences & Series

  • Remember: AP formula an = a + (n-1)d, GP formula an = a × r^(n-1)
  • Formula: AP sum Sn = n/2[first + last], GP sum Sn = a(r^n-1)/(r-1)
  • Trap: Always check if |r| < 1 before using GP infinite sum formula S∞ = a/(1-r)
  • Shortcut: Three terms in AP satisfy 2×middle = sum of other two
  • Trick: For HP problems, convert to AP using reciprocals first
  • Quick check: In GP, ratio of consecutive terms must be constant
  • Time-saver: Use Sn = n/2(first + last) for AP when both end terms are known
←Binomial TheoremLogarithms→
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