Quick Revision Notes

Class 11 Mathematics

Chapter: Sequences and Series (अनुक्रम तथा श्रेणी)

1. Introduction to Sequence and Series (अनुक्रम तथा श्रेणी)

• Sequence (अनुक्रम): An arrangement of numbers in a definite order according to some rule.
• Series (श्रेणी): The indicated sum of the terms of a sequence. If $a_1, a_2, a_3, \dots, a_n$ is a sequence, then the expression $a_1 + a_2 + a_3 + \dots + a_n$ is a series.
• Progression (प्रगति): Sequences whose terms follow certain specific patterns are called progressions.

2. Arithmetic Progression (AP) (समांतर श्रेणी)

An AP is a sequence in which each term after the first differs from the preceding term by a fixed number. This fixed number is called the common difference (सार्वअंतर, $d$).

• General Form: $a, a+d, a+2d, a+3d, \dots$
• $n$-th Term (General Term): $$a_n = a + (n - 1)d$$ (where $a$ = first term, $d$ = common difference, $n$ = number of terms)
• $n$-th Term from the End: $$l - (n - 1)d$$ (where $l$ = last term)
• Sum of First $n$ Terms ($S_n$): $$S_n = \frac{n}{2}[2a + (n - 1)d]$$ or $$S_n = \frac{n}{2}[a + l]$$
• Relation between $n$-th term and Sum of $n$ terms: $$a_n = S_n - S_{n-1}$$

3. Geometric Progression (GP) (गुणोत्तर श्रेणी)

A GP is a sequence in which each term is non-zero and the ratio of any term to its preceding term is always constant. This constant is called the common ratio (सार्व अनुपात, $r$).

• General Form: $a, ar, ar^2, ar^3, \dots$
• $n$-th Term (General Term): $$a_n = ar^{n-1}$$ (where $a$ = first term, $r$ = common ratio)
• Sum of First $n$ Terms ($S_n$): $$S_n = \frac{a(r^n - 1)}{r - 1} \quad (\text{if } r > 1)$$ $$S_n = \frac{a(1 - r^n)}{1 - r} \quad (\text{if } r < 1)$$ $$S_n = n \cdot a \quad (\text{if } r = 1)$$
• Sum of Infinite GP ($S_\infty$): $$S_\infty = \frac{a}{1 - r} \quad (\text{valid only when } |r| < 1 \text{ or } -1 < r < 1)$$

4. Means (माध्य)

• Arithmetic Mean (AM) (समांतर माध्य): If $a, A, b$ are in AP, then the Arithmetic Mean $A$ between $a$ and $b$ is: $$A = \frac{a + b}{2}$$ Inserting $n$ AMs ($A_1, A_2, \dots, A_n$) between $a$ and $b$: Common difference $d = \frac{b - a}{n + 1}$

• Geometric Mean (GM) (गुणोत्तर माध्य): If $a, G, b$ are in GP, then the Geometric Mean $G$ between $a$ and $b$ is: $$G = \sqrt{ab}$$ Inserting $n$ GMs ($G_1, G_2, \dots, G_n$) between $a$ and $b$: Common ratio $r = \left(\frac{b}{a}\right)^{\frac{1}{n+1}}$

5. Relationship between AM and GM (AM और GM में संबंध)

For any two positive real numbers $a$ and $b$: $$\text{AM} \ge \text{GM} \implies \frac{a + b}{2} \ge \sqrt{ab}$$ (Equality holds if and only if $a = b$)

6. Standard Summation Formulas (विशेष श्रेणियों के योग)

Let $n$ be a positive integer, then:

• Sum of first $n$ natural numbers: $$\sum n = 1 + 2 + 3 + \dots + n = \frac{n(n + 1)}{2}$$

• Sum of squares of first $n$ natural numbers: $$\sum n^2 = 1^2 + 2^2 + 3^2 + \dots + n^2 = \frac{n(n + 1)(2n + 1)}{6}$$

• Sum of cubes of first $n$ natural numbers: $$\sum n^3 = 1^3 + 2^3 + 3^3 + \dots + n^3 = \left[\frac{n(n + 1)}{2}\right]^2 = (\sum n)^2$$