Constant Sequence in Mathematics

definition of sequence in math

Definition of Constant Sequence:

  • A sequence {sn} is said to be a constant sequence if all terms of a sequence are the same.
  • A sequence {sn} defined by sn = a for all nN is called a constant sequence.
  • Formally a sequence {sn} is said to be a constant sequence if {s_{n + 1}} = {s_n} for all n \ge 1.

For example, the sequence {a} = {a, a, a, a, a,…} is a constant sequence, because all terms of it are the same. Its range is singleton set {a}, which is a finite set.

Hence, a constant sequence is always a finite sequence.

Example 1: The sequence {2} = {2, 2, 2, …} is a constant sequence, because all terms of this sequence are the same.

Example 2: The sequence {0} = {0, 0, 0, …} is a constant sequence, because all terms of this sequence are the same.

Example 3: The sequence {-1} = {-1, -1, -1, …} is a constant sequence, because all terms of this sequence are the same.

Example 4: The sequence {(-1)n} = {-1, 1, -1, …} is not a constant sequence, because all terms of this sequence are not the same.


अचर अनुक्रम (Constant Sequence) की परिभाषा:

  • यदि किसी अनुक्रम {sn} के सभी पद समान हों, तो उस अनुक्रम को एक अचर अनुक्रम कहते  हैं | 
  • सभी nN के लिए sn = a द्वारा परिभाषित अनुक्रम {sn}, एक अचर अनुक्रम कहलाता है।
  • औपचारिक रूप से, अनुक्रम {sn} को एक अचर अनुक्रम कहा जाता है यदि सभी n \ge 1 के लिए, {s_{n + 1}} = {s_n}

उदाहरण के लिए, अनुक्रम {a} = {a, a, a, a, a,…} एक अचर अनुक्रम है, क्योंकि इस अनुक्रम के सभी पद समान हैं। इसका परिसर एकल समुच्चय {a} है, जो एक परिमित समुच्चय है।

इसलिए, एक अचर अनुक्रम हमेशा एक परिमित अनुक्रम होता है।

उदाहरण 1: अनुक्रम {2} = {2, 2, 2, …} एक अचर अनुक्रम है, क्योंकि इस अनुक्रम के सभी पद समान हैं।

उदाहरण 2: अनुक्रम {0} = {0, 0, 0, …} एक अचर अनुक्रम है, क्योंकि इस अनुक्रम के सभी पद समान हैं।

उदाहरण 3: अनुक्रम {-1} = {-1, -1, -1, …} एक अचर अनुक्रम है, क्योंकि इस अनुक्रम के सभी पद समान हैं।

उदाहरण 4: अनुक्रम {(-1)n} = {-1, 1, -1, …} एक अचर अनुक्रम नहीं है, क्योंकि इस अनुक्रम के सभी पद समान नहीं हैं।



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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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