Sequence and Series (अनुक्रम और शृंखला) In Hindi
अनुक्रम (Sequence): संख्याओं का एक क्रमबद्ध समूह होता है, जिसमें प्रत्येक संख्या को एक विशिष्ट स्थान (term) दिया जाता है। अनुक्रम में, क्रमिक संख्याओं के बीच एक निश्चित संबंध होता है।
श्रृंखला (Series): अनुक्रम के सभी संख्याओं का योग होता है।
Sequence and Series In English
Sequence: A sequence is an ordered list of numbers, where each number is called a term or an element of the sequence. For example, the sequence 2, 4, 6, 8, 10 is an ascending sequence of even numbers.
Series: A series, on the other hand, is the sum of the terms of a sequence. It is the result of adding up all the numbers in the sequence. The series corresponding to the sequence 2, 4, 6, 8, 10 would be 2 + 4 + 6 + 8 + 10 = 30.
Types of Sequence and Series
1. Arithmetic Sequence: Is a sequence in which each term is obtained by adding a fixed constant value, called the common difference, to the preceding term. In other words, the difference between any two consecutive terms in an arithmetic sequence is the same. For example, 2, 5, 8, 11, 14 is an arithmetic sequence with a common difference of 3.
- Formula: a_n = a_1 + d(n-1)
2. Geometric Sequence: Is a sequence in which each term is obtained by multiplying the preceding term by a fixed constant value, called the common ratio. In other words, the ratio between any two consecutive terms in a geometric sequence is the same.
- Formula: a_n = a_1 x \( r^(n – 1) \)
3. Harmonic Sequence: Is a sequence of numbers in which the reciprocals of the terms form an arithmetic sequence. In other words, a harmonic sequence is formed by taking the reciprocal of each term in an arithmetic sequence.
- Formula: a_n = 1/n
4. Fibonacci Sequence: Is a series of numbers in which each number is the sum of the two preceding numbers. The sequence starts with 0 and 1, and each subsequent number is the sum of the previous two numbers. The first few terms of the Fibonacci sequence are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …
- Formula: Fn = Fn-1 + Fn-2
Difference Between Sequences and Series
| Attribute | Sequences | Series |
|---|---|---|
| Definition | An ordered list of numbers following a specific pattern or rule. | The sum of the terms of a sequence. |
| Representation | Can be represented by listing the terms or by a formula. | Represented as the sum of terms, often using sigma notation. |
| Example | 1, 2, 3, 4, 5, … (Arithmetic Sequence) | 1 + 2 + 3 + 4 + 5 + … (Arithmetic Series) |
| Types | Arithmetic, Geometric, Harmonic, Fibonacci, etc. | Arithmetic, Geometric, Telescoping, etc. |
| Notation | Usually denoted as \(a_1, a_2, a_3, \ldots\) or using a formula. | Denoted as \(S_n\) for the sum of \(n\) terms or using sigma notation (\(\sum\)). |
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