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

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

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Prove the following by the principle of mathematical induction: 1.3+2.4+3.5++(2n-1)(2n+1)=(n(4n^(2)+6n-1))/(3)

(1^(4))/(1.3)+(2^(4))/(3.5)+(3^(4))/(5.7)+......+(n^(4)) /((2n-1)(2n+1))=(n(4n^(2)+6n+5))/(48)+(n)/(16(2n+1))

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)

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

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

Statement -1: (1^(2))/(1.3)+(2^(2))/(3.5)+(3^(2))/(5.7)+ . . . .+(n^(2))/((2n-1)(2n+1))=(n(n+1))/(2(2n+1)) Statement -2: (1)/(1.3)+(1)/(3.5)+(1)/(5.7)+ . . . .+(1)/((2n-1)(2n+1))=(1)/(2n+1)

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

Find lim_(n rarr oo)((1.3.5...(2n-1)}(n+1)^(4)]+[n^(4)(1.3.5...(2n-1)) (2n+1)]

lim_(n rarr oo){(1)/(1.3)+(1)/(3.5)+(1)/(5.7)+....+(1)/((2n+1)(2n+3 ))