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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 n^th 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 {s_n} and {t_n} are said to be equal or identical if and only if n^th 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 {s_n} and {t_n} 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 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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Equality of Two Sequences

Introduction of Sequence:

Equality of two sequences:

अनुक्रम का परिचय (Introduction of Sequence):

दो अनुक्रमों की समानता (Equality of two sequences):

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Introduction of Sequence:

Equality of two sequences:

अनुक्रम का परिचय (Introduction of Sequence):

दो अनुक्रमों की समानता (Equality of two sequences):

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