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}.

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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.
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