Sequences and Series Class 11 Maths Chapter Notes

Key Terminology

Sequence

An ordered list of numbers, called terms, arranged according to a specific rule. The terms are denoted by a₁, a₂, a₃, ..., aₙ.

Series

The sum of the terms of a sequence. If a₁, a₂, a₃, ... is a sequence, then the series is a₁ + a₂ + a₃ + ...

Arithmetic Progression (AP)

A sequence in which the difference between any two consecutive terms is constant. This constant difference is called the common difference (d).

Geometric Progression (GP)

A sequence in which the ratio of any two consecutive terms is constant. This constant ratio is called the common ratio (r).

nth Term (aₙ)

The general term of a sequence, which is a formula that gives the value of the term at the nth position.

Arithmetic Mean (AM)

For two numbers a and b, the AM is (a+b)/2. The number A is the AM between a and b if a, A, b are in AP.

Geometric Mean (GM)

For two positive numbers a and b, the GM is √ab. The number G is the GM between a and b if a, G, b are in GP.

Harmonic Progression (HP)

A sequence is in HP if the reciprocals of its terms are in AP. It is not in the main CBSE syllabus but is a related concept.

Understanding Sequences vs. Series

A common point of confusion is the difference between a sequence and a series. Think of a sequence as an ordered list of numbers following a certain pattern or rule. For example, 2, 4, 6, 8, ... is a sequence where each term is obtained by adding 2 to the previous one. The order matters. If we change the order, it's a different sequence. A sequence can be finite (has a last term) or infinite (continues indefinitely).

A series, on the other hand, is the sum of the terms in a sequence. So, for the sequence above, the corresponding series would be 2 + 4 + 6 + 8 + .... We are no longer just listing the numbers; we are interested in their cumulative total. A progression is a special type of sequence where the terms are defined by a specific mathematical formula relating a term to its preceding terms. The most common progressions you'll study are Arithmetic Progression (AP), characterized by a common difference ('d'), and Geometric Progression (GP), characterized by a common ratio ('r'). Understanding this distinction is the first step to mastering the chapter.

Arithmetic Progression (AP) vs. Geometric Progression (GP)

Key Formulas and Properties

• {"point":"Nth term of an AP: aₙ = a + (n-1)d"}
• {"point":"Sum of n terms of an AP: Sₙ = n/2 [2a + (n-1)d]"}
• {"point":"Nth term of a GP: aₙ = arⁿ⁻¹"}
• {"point":"Sum of n terms of a GP: Sₙ = a(rⁿ-1)/(r-1), for r ≠ 1"}
• {"point":"Sum of an infinite GP: S∞ = a / (1-r), valid only when |r| < 1."}
• {"point":"Relationship between AM and GM: For any two positive numbers a and b, AM ≥ GM. That is, (a+b)/2 ≥ √ab."}
• {"point":"Sum of first n natural numbers: Σn = n(n+1)/2"}
• {"point":"Sum of squares of first n natural numbers: Σn² = n(n+1)(2n+1)/6"}
• {"point":"Sum of cubes of first n natural numbers: Σn³ = [n(n+1)/2]² = (Σn)²"}
• {"point":"To find 'd' in an AP: d = a₂ - a₁ . To find 'r' in a GP: r = a₂ / a₁."}

Worked Example

• {"title":"Arithmetic Progression Example","bodyMarkdown":"Question: For the AP: 3, 8, 13, 18, ..., find the 20th term (a₂₀) and the sum of the first 10 terms (S₁₀).\n\nSolution:\n1. Identify a and d:\n First term (a) = 3\n Common difference (d) = 8 - 3 = 5\n\n2. Find the 20th term (a₂₀):\n Using the formula aₙ = a + (n-1)d:\n a₂₀ = 3 + (20-1) 5\n a₂₀ = 3 + 19 5 = 3 + 95 = 98\n\n3. Find the sum of the first 10 terms (S₁₀):\n Using the formula Sₙ = n/2 [2a + (n-1)d]:\n S₁₀ = 10/2 [23 + (10-1)5]\n S₁₀ = 5 [6 + 95] = 5 [6 + 45] = 5 51 = 255"}
• {"title":"Geometric Progression Example","bodyMarkdown":"Question: Find the sum of the first 8 terms of the GP: 2, 6, 18, ...\n\nSolution:\n1. Identify a and r:\n First term (a) = 2\n Common ratio (r) = 6 / 2 = 3\n\n2. Find the sum of the first 8 terms (S₈):\n Since r > 1, use the formula Sₙ = a(rⁿ-1)/(r-1):\n S₈ = 2(3⁸ - 1) / (3 - 1)\n S₈ = 2(6561 - 1) / 2\n S₈ = 6560"}

Exam Traps and Scoring Tips

A common mistake is confusing the formulas for AP and GP. Write them down at the start of the exam section to be safe. For infinite GP sum problems, ALWAYS check if |r| < 1. If it's not, the sum to infinity does not exist (or diverges), and you must state that. Also, when a question asks to find three terms in AP, it is often easier to assume them as a-d, a, a+d. Similarly, for three terms in GP, use a/r, a, ar. This simplifies calculations when the sum or product of the terms is given.

Practice Questions with Solutions

• An AP has its 5th term as 22 and 15th term as 62. What is the common difference? d = (a₁₅ - a₅) / (15-5) = (62-22)/10 = 40/10 = 4.
• What is the common ratio of the GP: 16, -8, 4, -2, ...? r = a₂ / a₁ = -8 / 16 = -1/2.
• Can an infinite GP with first term 10 and common ratio 2 have a finite sum? No. The sum to infinity for a GP exists only if the absolute value of the common ratio |r| is less than 1. Here, |2| is not less than 1.
• What is the sum of the first 20 natural numbers? Using Σn = n(n+1)/2, the sum is 20(21)/2 = 210.

Frequently Asked Questions

Frequently Asked Questions