Equality of Two Sequences

Lata Agarwal Mathpedia, Sequences
equality of two sequences

Introduction of Sequence:

A sequence is a set of numbers in a definite order of occurrence. It is denoted by {\left{ {{s_n}} \right}}, where {s_n} is the nth term of the sequence.

The order of terms is of importance in a sequence. Thus the sequence {\left{ {1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5}, \ldots } \right}} is different from the sequence {\left{ {\frac{1}{2},1,\frac{1}{4},\frac{1}{3},\frac{1}{5}, \ldots } \right}}, even though both contain the same terms. 

Equality of two sequences:

Two sequences {sn} and {tn} are said to be equal or identical if and only if nth terms of both sequences are equal, that is,

{s_n} = {t_n} for all n ∈ N.

♦ Example 1:

These two sequences {\left{ {{s_n}} \right}} = {\left{ {{n^2} - n} \right}} and {\left{ {{t_n}} \right}} = {\left{ {n\left( {n - 1} \right)} \right}} are equal, because,

{n^2} - n = n\left( {n - 1} \right) for all n ∈ N,

that is,

\left\{ {0,2,6,12,20, \ldots } \right\} = \left\{ {0,2,6,12,20, \ldots } \right\} for all n ∈ N.

And so,

{s_n} = {t_n} for all n ∈ N.

♦ Example 2:

These two sequences {\left{ {{s_n}} \right}} = {\left{ {1, - 1,\;1, - 1, \ldots } \right}} and {\left{ {{t_n}} \right}} = {\left{ { - 1,\;1, - 1,\;1, \ldots } \right}} are not equal, because,

{\left{ {{s_n}} \right}} = {\left{ {1, - 1,\;1, - 1,\;1, - 1, \ldots } \right}} = {\left{ {{{\left( { - 1} \right)}^{n + 1}}} \right}},

and,

{\left{ {{t_n}} \right}} = {\left{ { - 1,\;1, - 1,\;1, - 1,\;1 \ldots } \right}} = {\left{ {{{\left( { - 1} \right)}^n}} \right}}

And,

{\left( { - 1} \right)^{n + 1}} \ne {\left( { - 1} \right)^n},

implies that,

{s_n} \ne {t_n} for all n ∈ N.

अनुक्रम का परिचय (Introduction of Sequence):

अनुक्रम किसी घटना के घटित होने के एक निश्चित क्रम में संख्याओं का समुच्चय है। इसे {\left{ {{s_n}} \right}} द्वारा दर्शाया जाता है, जहां {s_n} अनुक्रम का n-वाँ पद है।

अनुक्रम में ‘क्रम’ का बहुत महत्व है। इसलिए अनुक्रम {\left{ {1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5}, \ldots } \right}}, अनुक्रम {\left{ {\frac{1}{2},1,\frac{1}{4},\frac{1}{3},\frac{1}{5}, \ldots } \right}} से अलग है, अर्थात दोनों अनुक्रम अलग-अलग है भले ही दोनों में समान पद हों।

दो अनुक्रमों की समानता (Equality of two sequences):

दो अनुक्रम {sn} and {tn} को समान या समरूप कहा जाता है यदि और केवल यदि दोनों अनुक्रमों के n-वें पद समान हों, अर्थात्, सभी n ∈ N के लिए,

{s_n} = {t_n}.

 

♦ उदाहरण 1:

ये दोनों अनुक्रम {\left{ {{s_n}} \right}} = {\left{ {{n^2} - n} \right}} और {\left{ {{t_n}} \right}} = {\left{ {n\left( {n - 1} \right)} \right}} समान या बराबर हैं, क्योंकि, सभी n ∈ N के लिए,

{n^2} - n = n\left( {n - 1} \right), अर्थात,

\left\{ {0,2,6,12,20, \ldots } \right\} = \left\{ {0,2,6,12,20, \ldots } \right\}.

इसलिए, सभी n ∈ N के लिए,

{s_n} = {t_n}.

♦ उदाहरण 2:

ये दोनों अनुक्रम {\left{ {{s_n}} \right}} = {\left{ {1, - 1,\;1, - 1, \ldots } \right}} और {\left{ {{t_n}} \right}} = {\left{ { - 1,\;1, - 1,\;1, \ldots } \right}} समान नहीं हैं, क्योंकि,

{\left{ {{s_n}} \right}} = {\left{ {1, - 1,\;1, - 1,\;1, - 1, \ldots } \right}} = {\left{ {{{\left( { - 1} \right)}^{n + 1}}} \right}},

तथा,

{\left{ {{t_n}} \right}} = {\left{ { - 1,\;1, - 1,\;1, - 1,\;1 \ldots } \right}} = {\left{ {{{\left( { - 1} \right)}^n}} \right}}

और,

{\left( { - 1} \right)^{n + 1}} \ne {\left( { - 1} \right)^n},

इसका मतलब है कि सभी n ∈ N के लिए,

{s_n} \ne {t_n}.

Copyrighted Material © 2019 - 2024 Prinsli.com - All rights reserved

All content on this website is copyrighted. It is prohibited to copy, publish or distribute the content and images of this website through any website, book, newspaper, software, videos, YouTube Channel or any other medium without written permission. You are not authorized to alter, obscure or remove any proprietary information, copyright or logo from this Website in any way. If any of these rules are violated, it will be strongly protested and legal action will be taken.

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.
Website Facebook LinkedIn

Related Articles

bounded below, bounded above, mathematics, bounded sequence example
Sequences

Bounded Sequence in Mathematics

Lata Agarwal Sequences

Bounded Below Sequence – A sequence {} is said to be bounded below if the range set of {} is bounded below. A sequence is bounded below if there exists a fixed real number , […]

Sequence in hindi
Mathpedia

Sequence in hindi – अनुक्रम: परिचय, परिभाषा, निरूपण, उदाहरण

Lata Agarwal Mathpedia, Sequences

Sequence in hindi: अनुक्रम: परिचय, परिभाषा, निरूपण, उदाहरण, वास्तविक संख्याओं के अनुक्रम (Sequences of Real Numbers)… […]

the limit of a sequence, definition and example with solution, examples of sequences
Mathpedia

Limit of a Sequence: Definition and Examples

Lata Agarwal Mathpedia, Sequences

Limit of a Sequence: If a sequence {} tends to a limit , then we say that is the limit of a sequence {}, and write . In other words, “A real number is the […]

1 Comment

Leave a Reply

Your email address will not be published.


*