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Basic Operations on Sequences

Basic Operations on Sequences:

Here, we will introduce the basic operations for sequences. We know that the operations of addition, subtraction, multiplication, and division can all be defined using real numbers. These operations can easily be applied to a set of sequences. These operations are extended in a term-by-term manner. Let’s look at the definitions for each operation on sequences.

Basic Operations on Sequences:

Let {s_n} and {t_n} be two sequences.

1. Addition of Sequences:

The sequence having n^th terms ${s_n} + {t_n}$ is called the sum (addition) of the sequences {s_n} and {t_n}, and it is denoted by { $\left{ {{s_n} + {t_n}} \right}$ }. That is,

$\left\{ {{s_n}} \right\} + \left\{ {{t_n}} \right\} = \left\{ {{s_n} + {t_n}} \right\}$ .

2. Difference of Sequences:

The sequence having n^th terms ${s_n} - {t_n}$ is called the difference of the sequences {s_n} and {t_n}, and it is denoted by { $\left{ {{s_n} - {t_n}} \right}$ }. That is,

$\left\{ {{s_n}} \right\} - \left\{ {{t_n}} \right\} = \left\{ {{s_n} - {t_n}} \right\}$ .

3. Multiplication of Sequences:

The sequence having n^th terms ${s_n}{t_n}$ is called the product (multiplication) of the sequences {s_n} and {t_n}, and it is denoted by { $\left{ {{s_n}{t_n}} \right}$ }. That is,

$\left\{ {{s_n}} \right\} \cdot \left\{ {{t_n}} \right\} = \left\{ {{s_n}{t_n}} \right\}$ .

4. Reciprocal of the Sequence:

If ${t_n} \ne 0$ for all n∈N, then the sequence having n^th terms $\frac{1}{{{t_n}}}$ is called the reciprocal of the sequence $\left{ {{t_n}} \right}$ , and it is denoted by { $\left{ {\frac{1}{{{t_n}}}} \right}$ }.

5. Division of Sequences:

The sequence having n^th terms ${\frac{{{s_n}}}{{{t_n}}}}$ is called the division of the sequences {s_n} and {t_n}, and it is denoted by { $\left{ {\frac{{{s_n}}}{{{t_n}}}} \right}$ }. That is,

$\frac{{\left\{ {{s_n}} \right\}}}{{\left\{ {{t_n}} \right\}}} = \left\{ {\frac{{{s_n}}}{{{t_n}}}} \right\}$ .

6. Scalar Multiplication of a Sequence:

If c∈R, then the sequence having n^th terms ${c{s_n}}$ is called the scalar multiple of the sequence $\left{ {{s_n}} \right}$ , and it is denoted by { $\left{ {c{s_n}} \right}$ }. That is,

$c\left\{ {{s_n}} \right\} = \left\{ {c{s_n}} \right\}$ .

अनुक्रमों पर मूल संक्रियाएं (Basic Operations on Sequences):

यहां, हम अनुक्रमों के लिए मूल संक्रियाओं (basic operations) का परिचय देंगे। हम जानते हैं कि वास्तविक संख्याओं का उपयोग करके जोड़, घटाव, गुणा और भाग सभी को परिभाषित किया जा सकता है। इन संक्रियाओं को आसानी से अनुक्रमों के एक समुच्चय पर लागू किया जा सकता है। इन संक्रियाओं को टर्म-बाय-टर्म (term-by-term) तरीके से विस्तारित किया जाता है। आइए, अनुक्रमों पर प्रत्येक संक्रिया की परिभाषाओं को देखें।

अनुक्रमों पर मूल संक्रियाएं (Basic Operations on Sequences):

मान लीजिए {s_n} और {t_n} दो अनुक्रम हैं।

1. अनुक्रमों का जोड़ (Addition of Sequences):

nवें पदों वाले अनुक्रम ${s_n} + {t_n}$ को अनुक्रमों {s_n} और {t_n} का योग (जोड़) कहा जाता है, और इसे { $\left{ {{s_n} + {t_n}} \right}$ } द्वारा दर्शाया जाता है। अर्थात्,

$\left\{ {{s_n}} \right\} + \left\{ {{t_n}} \right\} = \left\{ {{s_n} + {t_n}} \right\}$ .

2. अनुक्रमों का अंतर (Difference of Sequences):

nवें पदों वाले अनुक्रम ${s_n} - {t_n}$ को अनुक्रमों {s_n} और {t_n} का अंतर (घटाव) कहा जाता है, और इसे { $\left{ {{s_n} - {t_n}} \right}$ } द्वारा दर्शाया जाता है। अर्थात्,

$\left\{ {{s_n}} \right\} - \left\{ {{t_n}} \right\} = \left\{ {{s_n} - {t_n}} \right\}$ .

3. अनुक्रमों का गुणन (Multiplication of Sequences):

nवें पदों वाले अनुक्रम ${s_n}{t_n}$ को अनुक्रमों {s_n} और {t_n} का गुणनफल (गुणा) कहा जाता है, और इसे { $\left{ {{s_n}{t_n}} \right}$ } द्वारा दर्शाया जाता है। अर्थात्,

$\left\{ {{s_n}} \right\} \cdot \left\{ {{t_n}} \right\} = \left\{ {{s_n}{t_n}} \right\}$ .

4. अनुक्रम का व्युत्क्रम या गुणन प्रतिलोम (Reciprocal of the Sequence):

यदि सभी n∈N के लिए ${t_n} \ne 0$ , तो nवें पदों वाले अनुक्रम $\frac{1}{{{t_n}}}$ को अनुक्रम $\left{ {{t_n}} \right}$ का व्युत्क्रम कहा जाता है, और इसे { $\left{ {\frac{1}{{{t_n}}}} \right}$ } द्वारा दर्शाया जाता है।

5. अनुक्रमों का विभाजन (Division of Sequences):

nवें पदों वाले अनुक्रम ${\frac{{{s_n}}}{{{t_n}}}}$ को अनुक्रमों {s_n} और {t_n} का विभाजन (भाग) कहा जाता है, और इसे { $\left{ {\frac{{{s_n}}}{{{t_n}}}} \right}$ } द्वारा दर्शाया जाता है। अर्थात्,

$\frac{{\left\{ {{s_n}} \right\}}}{{\left\{ {{t_n}} \right\}}} = \left\{ {\frac{{{s_n}}}{{{t_n}}}} \right\}$ .

6. अनुक्रम का अदिश गुणन (Scalar Multiplication of a Sequence):

यदि c ∈ R, तो nवें पदों वाले अनुक्रम ${c{s_n}}$ को अनुक्रम $\left{ {{s_n}} \right}$ का अदिश गुणज कहा जाता है, और इसे { $\left{ {c{s_n}} \right}$ } द्वारा दर्शाया जाता है। अर्थात्,

$c\left\{ {{s_n}} \right\} = \left\{ {c{s_n}} \right\}$ .

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About Lata Agarwal 268 Articles

M.Phil in Mathematics, skilled in MS Office, MathType, Ti-83, Internet, etc., and Teaching with strong education professional. Passionate teacher and loves math. Worked as a Assistant Professor for BBA, BCA, BSC(CS & IT), BE, etc. Also, experienced SME (Mathematics) with a demonstrated history of working in the internet industry. Provide the well explained detailed solutions in step-by-step format for different branches of US mathematics textbooks.

Basic Operations on Sequences

Basic Operations on Sequences:

Basic Operations on Sequences:

1. Addition of Sequences:

2. Difference of Sequences:

3. Multiplication of Sequences:

4. Reciprocal of the Sequence:

5. Division of Sequences:

6. Scalar Multiplication of a Sequence:

अनुक्रमों पर मूल संक्रियाएं (Basic Operations on Sequences):

अनुक्रमों पर मूल संक्रियाएं (Basic Operations on Sequences):

1. अनुक्रमों का जोड़ (Addition of Sequences):

2. अनुक्रमों का अंतर (Difference of Sequences):

3. अनुक्रमों का गुणन (Multiplication of Sequences):

4. अनुक्रम का व्युत्क्रम या गुणन प्रतिलोम (Reciprocal of the Sequence):

5. अनुक्रमों का विभाजन (Division of Sequences):

6. अनुक्रम का अदिश गुणन (Scalar Multiplication of a Sequence):

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Basic Operations on Sequences:

Basic Operations on Sequences:

1. Addition of Sequences:

2. Difference of Sequences:

3. Multiplication of Sequences:

4. Reciprocal of the Sequence:

5. Division of Sequences:

6. Scalar Multiplication of a Sequence:

अनुक्रमों पर मूल संक्रियाएं (Basic Operations on Sequences):

अनुक्रमों पर मूल संक्रियाएं (Basic Operations on Sequences):

1. अनुक्रमों का जोड़ (Addition of Sequences):

2. अनुक्रमों का अंतर (Difference of Sequences):

3. अनुक्रमों का गुणन (Multiplication of Sequences):

4. अनुक्रम का व्युत्क्रम या गुणन प्रतिलोम (Reciprocal of the Sequence):

5. अनुक्रमों का विभाजन (Division of Sequences):

6. अनुक्रम का अदिश गुणन (Scalar Multiplication of a Sequence):

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