ગણિતીય અનુમાનનો સિદ્ધાંત

ગણિત ધોરણ 11 સેમિસ્ટર 2

ગણિત

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ગણિતીય અનુમાનનો સિદ્ધાંત

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

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

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

અહ ીં space comma space straight P open parentheses straight n close parentheses space colon space straight a plus open parentheses straight a plus straight d close parentheses plus open parentheses straight a plus 2 straight d close parentheses plus open parentheses straight a plus 3 straight d close parentheses plus.... plus open square brackets straight a plus open parentheses straight n minus straight a close parentheses straight d close square brackets space equals space straight n over 2 open square brackets 2 straight a plus open parentheses straight n minus 1 close parentheses straight d close square brackets comma space straight n space element of space straight N

હવે, n = 1 માટે

L.H.S. = a અને R.H.S. =

1 half open square brackets 8 a plus open parentheses 1 minus 1 close parentheses d close square brackets space equals space a

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

માટે, P (1) સત્ય છે.

therefore space straight a plus open parentheses straight a plus straight d close parentheses plus open parentheses straight a plus 2 straight d close parentheses plus open parentheses straight a plus 3 straight d close parentheses plus.... plus space open square brackets straight a plus open parentheses straight k minus 1 close parentheses straight d close square brackets space equals space straight k over 2 open square brackets 2 straight a plus open parentheses straight k minus 1 close parentheses straight d close square brackets 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,

L. H. S. space equals space a plus open parentheses a plus d close parentheses plus open parentheses a plus 2 d close parentheses plus open parentheses a plus 3 d close parentheses plus space... space plus open square brackets a plus open parentheses k minus 1 close parentheses d close square brackets plus open parentheses a minus k d close parentheses space space space space space space space space space space space space space space equals space k over 2 open square brackets 2 a plus open parentheses k minus 1 close parentheses d close square brackets plus open parentheses a plus k d close parentheses space space space space space open square brackets because space left parenthesis 1 right parenthesis space space પરથ ી space close square brackets space space space space space space space space space space space space space space equals space k a plus k over 2 open parentheses k minus 1 close parentheses d plus a plus k d space space space space space space space space space space space space space space equals space k a plus a plus k over 2 open parentheses k minus 1 close parentheses d plus k d

equals space open parentheses straight k plus 1 close parentheses straight a plus fraction numerator straight k squared straight d over denominator 2 end fraction minus kd over 2 plus kd equals open parentheses straight k plus 1 close parentheses straight a plus fraction numerator straight k squared straight d over denominator 2 end fraction plus kd over 2 equals space fraction numerator straight k plus 1 over denominator 2 end fraction times 2 straight a plus fraction numerator straight k plus 1 over denominator 2 end fraction times kd equals space fraction numerator straight k plus 1 over denominator 2 end fraction open square brackets 2 a plus k d close square brackets equals space fraction numerator k plus 1 over denominator 2 end fraction open square brackets 2 a plus open parentheses open parentheses k plus 1 close parentheses minus 1 close parentheses d close square brackets space equals space R. H. S.

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

સત્ય છે.

therefore space straight P open parentheses straight k close parentheses

સત્ય છે.

rightwards double arrow space straight P open parentheses straight k plus 1 close parentheses

સત્ય છે.

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

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