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ગણિતીય અનુમાનનો સિદ્ધાંત

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ગણિત ધોરણ 11 સેમિસ્ટર 2

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ગણિત

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1 plus 5 plus 9 plus.... plus open parentheses 4 n minus 3 close parentheses space equals space n open parentheses 2 n minus 1 close parentheses

અહીં, P (n) : 1 plus 5 plus 9 plus.... plus open parentheses 4 n minus 3 close parentheses space equals space n open parentheses 2 n minus 1 close parentheses comma space n space element of space N

હવે, n = 1 માટે,

straight L. straight H. straight S. space equals space 1 space અન ે space straight R. straight H. straight S. space equals space 1 space open parentheses 2 space times 1 minus 1 close parentheses space equals space 1

therefore space straight L. straight H. straight S. space equals space straight R. straight H. straight S.


therefore space straight P open parentheses 1 close parentheses સત્ય છે. 

therefore space 1 plus 5 plus 9 plus.... plus open parentheses 4 straight k minus 3 close parentheses space equals space straight k open parentheses 2 straight k minus 1 close parentheses space comma space straight k space element of space straight N space........ space left parenthesis 1 right parenthesis

હવે, P ( k + 1) સત્ય વતાવવા માટે n = k + 1 લેતાં,

straight L. straight H. straight S. space equals space 1 plus 5 plus 9 plus.... plus open parentheses 4 straight k minus 3 close parentheses plus open parentheses 4 open parentheses straight k plus 1 close parentheses minus 3 close parentheses

space space space space space space space space space space space space space equals space straight k open parentheses 2 straight k minus 1 close parentheses plus open parentheses 4 straight k plus 1 close parentheses space space open square brackets because space open parentheses 1 close parentheses space પરથ ી close square brackets

space space space space space space space space space space space space space equals space 2 straight k squared plus 3 straight k plus 1

space space space space space space space space space space space space space equals space open parentheses straight k plus 1 close parentheses open parentheses 2 straight k plus 1 close parentheses equals open parentheses straight k plus 1 close parentheses open parentheses 2 open parentheses straight k plus 1 close parentheses minus 1 close parentheses equals space straight R. straight H. straight S.




P ( k + 1 ) સત્ય છે.

P ( k ) સત્ય છે.rightwards double arrow P ( k + 1 ) સત્ય છે.

વળી, P (1) પણ સત્ય છે.

ગણિતીય અનુમાનના સિદ્ધાંતથી P(n), દરેક straight n space element of space straight N માટે સત્ય છે.


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