CBSE Class 11 Maths: Sequences and Series - Exercise 9.1 Deep Dive

Understanding Sequences: The Building Blocks

Sequence

A sequence is an arrangement of numbers in a definite order according to some rule. The numbers appearing in a sequence are called its terms. It's essentially a function whose domain is the set of natural numbers (1, 2, 3, ...) or some subset thereof.

Terms of a Sequence

The individual numbers in a sequence are called its terms. We generally denote the first term as a₁, the second term as a₂, the nth term as a_n, and so on. The subscript indicates the position of the term in the sequence.

Finite Sequence

A sequence containing a limited number of terms is called a finite sequence. For example, the sequence of even numbers less than 10 (2, 4, 6, 8) is a finite sequence with 4 terms.

Infinite Sequence

A sequence that has an unlimited number of terms is called an infinite sequence. The sequence of all natural numbers (1, 2, 3, ...) is an infinite sequence.

General Term (nth term)

The general term, often denoted as a_n or t_n, is a formula that allows you to find any term of the sequence by substituting the term's position (n) into the formula. It's the rule that governs the sequence.

How to Generate Terms from a Sequence's General Term

One of the fundamental skills you'll develop in Exercise 9.1 is the ability to generate the terms of a sequence when its general term (or nth term) is given. The general term, often expressed as an algebraic formula involving 'n', acts as a blueprint for every number in the sequence. The variable 'n' always represents the position of the term in the sequence, and it typically takes positive integer values starting from 1 (i.e., n = 1, 2, 3, 4, ...).

To find a specific term, say the 5th term, you simply substitute 'n = 5' into the formula for the general term. If you need the first few terms, you'll substitute n = 1, then n = 2, then n = 3, and so on, up to the desired number of terms. For instance, if the general term is given by a_n = 2n + 1, to find the first term, we put n=1, which gives a_1 = 2(1) + 1 = 3. For the second term, n=2, so a_2 = 2(2) + 1 = 5. This process is repeated to generate as many terms as required.

It's crucial to pay close attention to the details of the general term, especially signs, exponents, and fractions. A common variation might involve (-1)^n or (-1)^(n+1), which alternates the sign of the terms. When 'n' is in the exponent or denominator, careful calculation is essential. Remember, understanding this mechanism is key to unlocking the entire chapter, as sequences form the basis for series, which are sums of the terms of a sequence.

Worked Examples: Finding Sequence Terms

• Example 1: Write the first five terms of the sequence whose nth term is given by a_n = 2n + 5. Step 1: Understand the general term. The general term is a_n = 2n + 5. We need to find the terms for n = 1, 2, 3, 4, 5. Step 2: Substitute n=1. a₁ = 2(1) + 5 = 2 + 5 = 7 Step 3: Substitute n=2. a₂ = 2(2) + 5 = 4 + 5 = 9 Step 4: Substitute n=3. a₃ = 2(3) + 5 = 6 + 5 = 11 Step 5: Substitute n=4. a₄ = 2(4) + 5 = 8 + 5 = 13 Step 6: Substitute n=5. a₅ = 2(5) + 5 = 10 + 5 = 15 Final Answer: The first five terms are 7, 9, 11, 13, 15.
• Example 2: Find the first four terms of the sequence defined by a_n = n(n + 2). Step 1: Identify the general term. The general term is a_n = n(n + 2). We need terms for n = 1, 2, 3, 4. Step 2: Calculate for n=1. a₁ = 1(1 + 2) = 1(3) = 3 Step 3: Calculate for n=2. a₂ = 2(2 + 2) = 2(4) = 8 Step 4: Calculate for n=3. a₃ = 3(3 + 2) = 3(5) = 15 Step 5: Calculate for n=4. a₄ = 4(4 + 2) = 4(6) = 24 Final Answer: The first four terms are 3, 8, 15, 24.
• Example 3: Write the first three terms of the sequence a_n = (-1)^(n-1) n^3. Step 1: Note the general term. a_n = (-1)^(n-1) n^3. We need terms for n = 1, 2, 3. Step 2: Calculate for n=1. a₁ = (-1)^(1-1) 1^3 = (-1)^0 1 = 1 1 = 1 Step 3: Calculate for n=2. a₂ = (-1)^(2-1) 2^3 = (-1)^1 8 = -1 8 = -8 Step 4: Calculate for n=3. a₃ = (-1)^(3-1) 3^3 = (-1)^2 27 = 1 * 27 = 27 Final Answer: The first three terms are 1, -8, 27.

Exam Tip: Avoiding Common Pitfalls

When working with sequences and their general terms, especially in Exercise 9.1, be mindful of these common mistakes:

1. Incorrect 'n' value: Always remember that 'n' usually starts from 1 (for the first term) and represents a positive integer (1, 2, 3, ...). Do not start with n=0 unless explicitly stated in the problem.
2. Careless Calculations: Double-check your arithmetic, especially with fractions, negative signs, and exponents. A small error in substitution or calculation can lead to an entirely wrong term.
3. Understanding (-1)^n: When the general term involves (-1)^n or (-1)^(n+1), remember that it dictates an alternating sign. If n is even, (-1)^n is positive. If n is odd, (-1)^n is negative. The n+1 or n-1 in the exponent will simply shift this pattern.
4. Reading the Question: Pay attention to how many terms are asked (e.g., first five terms, first four terms). Don't stop prematurely or calculate more terms than necessary. Ensure your final answer lists the terms clearly, separated by commas.

Practice Questions with Solutions

• Q: Write the first five terms of the sequence whose nth term is a_n = (n^2 + 1) / (n + 1). A: Step 1: Understand the general term. We need terms for n = 1, 2, 3, 4, 5. Step 2: For n=1, a₁ = (1^2 + 1) / (1 + 1) = (1 + 1) / 2 = 2 / 2 = 1. Step 3: For n=2, a₂ = (2^2 + 1) / (2 + 1) = (4 + 1) / 3 = 5 / 3. Step 4: For n=3, a₃ = (3^2 + 1) / (3 + 1) = (9 + 1) / 4 = 10 / 4 = 5 / 2. Step 5: For n=4, a₄ = (4^2 + 1) / (4 + 1) = (16 + 1) / 5 = 17 / 5. Step 6: For n=5, a₅ = (5^2 + 1) / (5 + 1) = (25 + 1) / 6 = 26 / 6 = 13 / 3. Final answer: The first five terms are 1, 5/3, 5/2, 17/5, 13/3.
• Q: Find the first four terms of the sequence whose nth term is a_n = (n - 1) 2^n. A: Step 1: General term is a_n = (n - 1) 2^n. We need terms for n = 1, 2, 3, 4. Step 2: For n=1, a₁ = (1 - 1) 2^1 = 0 2 = 0. Step 3: For n=2, a₂ = (2 - 1) 2^2 = 1 4 = 4. Step 4: For n=3, a₃ = (3 - 1) 2^3 = 2 8 = 16. Step 5: For n=4, a₄ = (4 - 1) 2^4 = 3 16 = 48. Final answer: The first four terms are 0, 4, 16, 48.
• Q: Write the first three terms of the sequence defined by a_n = (n^2) / (n + 2). A: Step 1: General term is a_n = (n^2) / (n + 2). We need terms for n = 1, 2, 3. Step 2: For n=1, a₁ = (1^2) / (1 + 2) = 1 / 3. Step 3: For n=2, a₂ = (2^2) / (2 + 2) = 4 / 4 = 1. Step 4: For n=3, a₃ = (3^2) / (3 + 2) = 9 / 5. Final answer: The first three terms are 1/3, 1, 9/5.
• Q: Determine the first three terms of the sequence if a_n = (-1)^n / (n + 1). A: Step 1: General term is a_n = (-1)^n / (n + 1). We need terms for n = 1, 2, 3. Step 2: For n=1, a₁ = (-1)^1 / (1 + 1) = -1 / 2. Step 3: For n=2, a₂ = (-1)^2 / (2 + 1) = 1 / 3. Step 4: For n=3, a₃ = (-1)^3 / (3 + 1) = -1 / 4. Final answer: The first three terms are -1/2, 1/3, -1/4.

Frequently Asked Questions