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" प्रक्क.23.गणितीय आगमन सिद्धांत से सिद्...

" प्रक्क.23.गणितीय आगमन सिद्धांत से सिद्ध कीजिए कि "1+3+5+.................-(2n-1)=n^(2),AA n in N

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Prove 1 + 5 + 9 + ... + (4n - 3) = n(2n - 1), AA n in N

1.2+2.3+3.4+.........+n(n+1)=(1)/(3)n(n+1)(n+3)

1.3+3.5+5.7+......+(2n-1)(2n+1)=(n(4n^(2)+6n-1))/(3)

Prove by PMI that 1.2+2.3+3.4+....+n(n+1)=((n)(n+1)(n+2))/(3),AA n in N

The A.M.of the observations 1.3.5,3.5.7,5.7.9,......,(2n-1)(2n+1)(2n+3) is (AA n in N)

1.3+2.3^(2)+3.3^(3)+............+n.3^(n)=((2n-1)3^(n+1)+3 )/(4)

Prove that ((2n+1)!)/(n !)=2^n{1. 3. 5 .........(2n-1)(2n+1)}

1.2+2.3+3.4+…………..+n(n+1)=n/3(n+1)(n+2) forall n in N.

Given f(1)=2 and f(n+1)=(f(n)-1)/(f(n)+1)AA n in N then

lim_(n rarr oo) (1.2 +2.3+3.4+ .....+n(n+1))/n^(3)=