Basic Operations on Sequences

definition of sequence in math

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 {sn} and {tn} be two sequences.

1. Addition of Sequences:

The sequence having nth terms {s_n} + {t_n} is called the sum (addition) of the sequences {sn} and {tn}, 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 nth terms {s_n} - {t_n} is called the difference of the sequences {sn} and {tn}, 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 nth terms {s_n}{t_n} is called the product (multiplication) of the sequences {sn} and {tn}, 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 nтИИN, then the sequence having nth 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 nth terms {\frac{{{s_n}}}{{{t_n}}}} is called the division of the sequences {sn} and {tn}, 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 cтИИR, then the sequence having nth 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\}.


рдЕрдиреБрдХреНрд░рдореЛрдВ рдкрд░ рдореВрд▓ рд╕рдВрдХреНрд░рд┐рдпрд╛рдПрдВ (Basic Operations on Sequences):

рдпрд╣рд╛рдВ, рд╣рдо рдЕрдиреБрдХреНрд░рдореЛрдВ рдХреЗ рд▓рд┐рдП рдореВрд▓ рд╕рдВрдХреНрд░рд┐рдпрд╛рдУрдВ (basic operations) рдХрд╛ рдкрд░рд┐рдЪрдп рджреЗрдВрдЧреЗред рд╣рдо рдЬрд╛рдирддреЗ рд╣реИрдВ рдХрд┐ рд╡рд╛рд╕реНрддрд╡рд┐рдХ рд╕рдВрдЦреНрдпрд╛рдУрдВ рдХрд╛ рдЙрдкрдпреЛрдЧ рдХрд░рдХреЗ рдЬреЛрдбрд╝, рдШрдЯрд╛рд╡, рдЧреБрдгрд╛ рдФрд░ рднрд╛рдЧ рд╕рднреА рдХреЛ рдкрд░рд┐рднрд╛рд╖рд┐рдд рдХрд┐рдпрд╛ рдЬрд╛ рд╕рдХрддрд╛ рд╣реИред рдЗрди рд╕рдВрдХреНрд░рд┐рдпрд╛рдУрдВ рдХреЛ рдЖрд╕рд╛рдиреА рд╕реЗ рдЕрдиреБрдХреНрд░рдореЛрдВ рдХреЗ рдПрдХ рд╕рдореБрдЪреНрдЪрдп рдкрд░ рд▓рд╛рдЧреВ рдХрд┐рдпрд╛ рдЬрд╛ рд╕рдХрддрд╛ рд╣реИред рдЗрди рд╕рдВрдХреНрд░рд┐рдпрд╛рдУрдВ рдХреЛ рдЯрд░реНрдо-рдмрд╛рдп-рдЯрд░реНрдо (term-by-term) рддрд░реАрдХреЗ рд╕реЗ рд╡рд┐рд╕реНрддрд╛рд░рд┐рдд рдХрд┐рдпрд╛ рдЬрд╛рддрд╛ рд╣реИред рдЖрдЗрдП, рдЕрдиреБрдХреНрд░рдореЛрдВ рдкрд░ рдкреНрд░рддреНрдпреЗрдХ рд╕рдВрдХреНрд░рд┐рдпрд╛ рдХреА рдкрд░рд┐рднрд╛рд╖рд╛рдУрдВ рдХреЛ рджреЗрдЦреЗрдВред

рдЕрдиреБрдХреНрд░рдореЛрдВ рдкрд░ рдореВрд▓ рд╕рдВрдХреНрд░рд┐рдпрд╛рдПрдВ (Basic Operations on Sequences):

рдорд╛рди рд▓реАрдЬрд┐рдП {sn} рдФрд░ {tn} рджреЛ рдЕрдиреБрдХреНрд░рдо рд╣реИрдВред

1. рдЕрдиреБрдХреНрд░рдореЛрдВ рдХрд╛ рдЬреЛрдбрд╝ (Addition of Sequences):

nрд╡реЗрдВ рдкрджреЛрдВ рд╡рд╛рд▓реЗ рдЕрдиреБрдХреНрд░рдо {s_n} + {t_n} рдХреЛ рдЕрдиреБрдХреНрд░рдореЛрдВ {sn} рдФрд░ {tn} рдХрд╛ рдпреЛрдЧ (рдЬреЛрдбрд╝) рдХрд╣рд╛ рдЬрд╛рддрд╛ рд╣реИ, рдФрд░ рдЗрд╕реЗ {\left{ {{s_n} + {t_n}} \right}} рджреНрд╡рд╛рд░рд╛ рджрд░реНрд╢рд╛рдпрд╛ рдЬрд╛рддрд╛ рд╣реИред рдЕрд░реНрдерд╛рддреН,

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

2. рдЕрдиреБрдХреНрд░рдореЛрдВ рдХрд╛ рдЕрдВрддрд░ (Difference of Sequences):

nрд╡реЗрдВ рдкрджреЛрдВ рд╡рд╛рд▓реЗ рдЕрдиреБрдХреНрд░рдо {s_n} - {t_n} рдХреЛ рдЕрдиреБрдХреНрд░рдореЛрдВ {sn} рдФрд░ {tn} рдХрд╛ рдЕрдВрддрд░ (рдШрдЯрд╛рд╡) рдХрд╣рд╛ рдЬрд╛рддрд╛ рд╣реИ, рдФрд░ рдЗрд╕реЗ {\left{ {{s_n} - {t_n}} \right}} рджреНрд╡рд╛рд░рд╛ рджрд░реНрд╢рд╛рдпрд╛ рдЬрд╛рддрд╛ рд╣реИред рдЕрд░реНрдерд╛рддреН,

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

3. рдЕрдиреБрдХреНрд░рдореЛрдВ рдХрд╛ рдЧреБрдгрди (Multiplication of Sequences):

nрд╡реЗрдВ рдкрджреЛрдВ рд╡рд╛рд▓реЗ рдЕрдиреБрдХреНрд░рдо {s_n}{t_n} рдХреЛ рдЕрдиреБрдХреНрд░рдореЛрдВ {sn} рдФрд░ {tn} рдХрд╛ рдЧреБрдгрдирдлрд▓ (рдЧреБрдгрд╛) рдХрд╣рд╛ рдЬрд╛рддрд╛ рд╣реИ, рдФрд░ рдЗрд╕реЗ {\left{ {{s_n}{t_n}} \right}} рджреНрд╡рд╛рд░рд╛ рджрд░реНрд╢рд╛рдпрд╛ рдЬрд╛рддрд╛ рд╣реИред рдЕрд░реНрдерд╛рддреН,

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

4. рдЕрдиреБрдХреНрд░рдо рдХрд╛ рд╡реНрдпреБрддреНрдХреНрд░рдо рдпрд╛ рдЧреБрдгрди рдкреНрд░рддрд┐рд▓реЛрдо (Reciprocal of the Sequence):

рдпрджрд┐ рд╕рднреА nтИИN рдХреЗ рд▓рд┐рдП {t_n} \ne 0, рддреЛ nрд╡реЗрдВ рдкрджреЛрдВ рд╡рд╛рд▓реЗ рдЕрдиреБрдХреНрд░рдо \frac{1}{{{t_n}}} рдХреЛ рдЕрдиреБрдХреНрд░рдо \left{ {{t_n}} \right} рдХрд╛ рд╡реНрдпреБрддреНрдХреНрд░рдо рдХрд╣рд╛ рдЬрд╛рддрд╛ рд╣реИ, рдФрд░ рдЗрд╕реЗ {\left{ {\frac{1}{{{t_n}}}} \right}} рджреНрд╡рд╛рд░рд╛ рджрд░реНрд╢рд╛рдпрд╛ рдЬрд╛рддрд╛ рд╣реИред

5. рдЕрдиреБрдХреНрд░рдореЛрдВ рдХрд╛ рд╡рд┐рднрд╛рдЬрди (Division of Sequences):

nрд╡реЗрдВ рдкрджреЛрдВ рд╡рд╛рд▓реЗ рдЕрдиреБрдХреНрд░рдо {\frac{{{s_n}}}{{{t_n}}}} рдХреЛ рдЕрдиреБрдХреНрд░рдореЛрдВ {sn} рдФрд░ {tn} рдХрд╛ рд╡рд┐рднрд╛рдЬрди (рднрд╛рдЧ) рдХрд╣рд╛ рдЬрд╛рддрд╛ рд╣реИ, рдФрд░ рдЗрд╕реЗ {\left{ {\frac{{{s_n}}}{{{t_n}}}} \right}} рджреНрд╡рд╛рд░рд╛ рджрд░реНрд╢рд╛рдпрд╛ рдЬрд╛рддрд╛ рд╣реИред рдЕрд░реНрдерд╛рддреН,

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

6. рдЕрдиреБрдХреНрд░рдо рдХрд╛ рдЕрджрд┐рд╢ рдЧреБрдгрди (Scalar Multiplication of a Sequence):

рдпрджрд┐ c тИИ R, рддреЛ nрд╡реЗрдВ рдкрджреЛрдВ рд╡рд╛рд▓реЗ рдЕрдиреБрдХреНрд░рдо {c{s_n}} рдХреЛ рдЕрдиреБрдХреНрд░рдо \left{ {{s_n}} \right} рдХрд╛ рдЕрджрд┐рд╢ рдЧреБрдгрдЬ рдХрд╣рд╛ рдЬрд╛рддрд╛ рд╣реИ, рдФрд░ рдЗрд╕реЗ {\left{ {c{s_n}} \right}} рджреНрд╡рд╛рд░рд╛ рджрд░реНрд╢рд╛рдпрд╛ рдЬрд╛рддрд╛ рд╣реИред рдЕрд░реНрдерд╛рддреН,

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



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About Lata Agarwal 270 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.

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