Class 11 Mathematics (041) · Chapter 8 NotesSequences and Series

A sequence is an ordered list of numbers following a specific pattern or rule. Each number in the sequence is called a term. Sequences can be finite, containing a limited number of terms, or infinite, extending indefinitely. For example, the sequence of even numbers 2, 4, 6, 8, ... is infinite, while the sequence of a person's ancestors over 300 years with a 30-year generation gap is finite. Sequences can be defined explicitly by a formula for the nth term or recursively by a relationship between consecutive terms.

• A sequence is a list of numbers in a specific order
• Each number in the sequence is called a term
• Sequences can be finite or infinite

A series is the sum of the terms of a sequence. It is represented by the sigma notation, which indicates the summation of terms. For example, the series associated with the sequence a1, a2, a3, ... is written as Σa_k from k=1 to n. Series can be finite or infinite, depending on the nature of the sequence. The sum of a series is the result of adding all its terms. For instance, the sum of the series 1 + 3 + 5 + 7 is 16.

• A series is the sum of the terms of a sequence
• Series are represented using sigma notation
• Series can be finite or infinite

An arithmetic progression (AP) is a sequence where each term after the first is obtained by adding a constant difference to the preceding term. This constant difference is called the common difference, denoted by d. The nth term of an AP can be found using the formula an = a1 + (n-1)d, where a1 is the first term. For example, the sequence 2, 5, 8, 11, ... is an AP with a common difference of 3. The sum of the first n terms of an AP is given by Sn = n/2 (2a1 + (n-1)d).

• An AP is a sequence with a constant difference between consecutive terms
• The nth term of an AP is given by an = a1 + (n-1)d
• The sum of the first n terms of an AP is Sn = n/2 (2a1 + (n-1)d)

The arithmetic mean (AM) of two positive numbers a and b is the number (a + b)/2. It is the average of the two numbers and represents the middle value between them. For example, the AM of 4 and 10 is (4 + 10)/2 = 7. The AM is a measure of central tendency and is widely used in statistics and various applications. The relationship between AM and geometric mean (GM) is given by AM ≥ GM, with equality holding if and only if the two numbers are equal.

• The AM of two numbers a and b is (a + b)/2
• AM is the average of the two numbers
• AM ≥ GM for any two positive numbers

A geometric progression (GP) is a sequence where each term after the first is obtained by multiplying the preceding term by a constant ratio, denoted by r. The nth term of a GP is given by an = ar^(n-1), where a is the first term. For example, the sequence 3, 9, 27, 81, ... is a GP with a common ratio of 3. The sum of the first n terms of a GP is given by Sn = a(1 - r^n)/(1 - r) if r ≠ 1. If r = 1, the sum is Sn = n * a. GPs have important applications in various fields, including finance, physics, and computer science.

• A GP is a sequence with a constant ratio between consecutive terms
• The nth term of a GP is given by an = ar^(n-1)
• The sum of the first n terms of a GP is Sn = a(1 - r^n)/(1 - r) if r ≠ 1

The general term or nth term of a GP is given by the formula an = ar^(n-1). This formula allows us to find any term in the sequence without listing all the previous terms. For example, in the GP 5, 25, 125, ..., the 10th term can be found using a10 = 5 * 5^(10-1) = 5^10. The general term formula is derived from the definition of a GP, where each term is obtained by multiplying the previous term by the common ratio r. This formula is essential for solving problems involving GPs and for understanding the behavior of the sequence.

• The general term of a GP is an = ar^(n-1)
• This formula allows finding any term in the sequence
• The formula is derived from the definition of a GP

The sum of the first n terms of a GP is given by the formula Sn = a(1 - r^n)/(1 - r), where a is the first term, r is the common ratio, and n is the number of terms. This formula is valid when r ≠ 1. If r = 1, the sum simplifies to Sn = n * a, as all terms are equal. For example, the sum of the first 5 terms of the GP 1, 1/2, 1/4, 1/8, ... is S5 = 1(1 - (1/2)^5)/(1 - 1/2) = 1.875. The sum formula is derived by multiplying the sum by the common ratio and subtracting it from the original sum, leading to a telescoping series.

• The sum of the first n terms of a GP is Sn = a(1 - r^n)/(1 - r)
• If r = 1, the sum is Sn = n * a
• The formula is derived using a telescoping series