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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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ગણિતીય અનુમાનનો સિદ્ધાંત

1 squared space times space 2 space plus space 2 squared space times space 3 space plus space... space plus space straight n squared space open parentheses straight n plus 1 close parentheses space equals space fraction numerator straight n open parentheses straight n plus 1 close parentheses open parentheses straight n plus 2 close parentheses open parentheses 3 straight n plus 1 close parentheses over denominator 12 end fraction

માટે P (1) સત્ય છે.

ધારો કે, P(k) સત્ય છે.

therefore space fraction numerator 1 over denominator 1 times 4 end fraction plus fraction numerator 1 over denominator 4 times 7 end fraction plus fraction numerator 1 over denominator 7 times 10 end fraction plus.... plus fraction numerator 1 over denominator open parentheses 3 straight k minus 2 close parentheses open parentheses 3 straight k plus 1 close parentheses end fraction equals space fraction numerator straight k over denominator 3 straight k plus 1 end fraction comma space straight k space element of space straight N space....... left parenthesis 1 right parenthesis

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

straight L. straight H. straight S. space equals space fraction numerator 1 over denominator 1 times 4 end fraction plus fraction numerator 1 over denominator 4 times 7 end fraction plus fraction numerator 1 over denominator 7 times 10 end fraction plus.... plus fraction numerator 1 over denominator open parentheses 3 straight k minus 2 close parentheses open parentheses 3 straight k plus 1 close parentheses end fraction plus fraction numerator 1 over denominator open parentheses 3 open parentheses straight k plus 1 close parentheses minus 2 close parentheses open parentheses 3 open parentheses straight k plus 1 close parentheses plus 1 close parentheses end fraction space space space space space space space space space space space space space space equals space fraction numerator straight k over denominator 3 straight k plus 1 end fraction plus fraction numerator 1 over denominator open parentheses 3 open parentheses straight k plus 1 close parentheses minus 2 close parentheses open parentheses 3 open parentheses straight k plus 1 close parentheses plus 1 close parentheses end fraction space space space space open square brackets because space left parenthesis 1 right parenthesis space પરથ ી space close square brackets space space space space space space space space space space space space space equals space fraction numerator straight k over denominator 3 straight k plus 1 end fraction plus fraction numerator 1 over denominator open parentheses 3 straight k plus 1 close parentheses open parentheses 3 straight k plus 4 close parentheses end fraction space space space space space space space space space space space space equals space fraction numerator straight k open parentheses 3 straight k plus 4 close parentheses plus 1 over denominator open parentheses 3 straight k plus 1 close parentheses open parentheses 3 straight k plus 4 close parentheses end fraction space

equals space fraction numerator 3 straight k squared plus 4 straight k plus 1 over denominator open parentheses 3 straight k plus 1 close parentheses open parentheses 3 straight k plus 4 close parentheses end fraction equals space fraction numerator open parentheses 3 straight k plus 1 close parentheses open parentheses straight k plus 1 close parentheses over denominator open parentheses 3 straight k plus 1 close parentheses open parentheses 3 straight k plus 4 close parentheses end fraction equals space fraction numerator straight k plus 1 over denominator 3 straight k plus 4 end fraction equals space fraction numerator straight k plus 1 over denominator 3 open parentheses straight k plus 1 close parentheses plus 1 end fraction space equals space straight R. straight H. straight S.

therefore space straight P space open parentheses straight k plus 1 close parentheses

therefore space straight P open parentheses straight k close parentheses space સત ્ ય space છ ે. space rightwards double arrow space straight P space open parentheses straight k plus 1 close parentheses space સત ્ ય space છ ે.

વળી, P(1) પણ સત્ય છે.

તેથી ગણિતીય અનુમાનના સિદ્ધાંતથી P(n), દરેક

n space element of space N

માટે સત્ય છે.

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

1 times 2 times 3 space plus space 2 times 3 times 4 plus.... plus straight n open parentheses straight n plus 1 close parentheses open parentheses straight n plus 2 close parentheses space equals space fraction numerator straight n open parentheses straight n plus 1 close parentheses open parentheses straight n plus 2 close parentheses open parentheses straight n plus 3 close parentheses over denominator 4 end fraction

straight a plus open parentheses straight a plus straight d close parentheses plus open parentheses straight a plus 2 straight d close parentheses plus.... plus open parentheses straight a plus open parentheses straight n minus straight a close parentheses straight d close parentheses space equals space 1 half straight n open parentheses 2 straight a plus open parentheses straight n minus 1 close parentheses straight d close parentheses

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