How to evaluate 1(3) 3(5) 5(7) … (2n-1)(2n 1)

1(3) + 3(5) + 5(7) + … + (2n-1)(2n+1)

`lim_(n- gtoo)(1/(1. 3)+1/(3. 5)+................+1/((2n-1)(2n+1)))`

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p(2n-1):p(2n+1,n-1)=5:3

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

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`(1)/(3.5)+(1)/(5.7)+(1)/(7.9)+......` `+(1)/((2n+1)(2n+3))=(n)/(3(2n+3))`

If `((243)^(n/5) 3^(2n+1))/(9^n 3^(n-1)) = x` , then the value of x is

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Fnd the value: `((243)^(n/5) 3^(2n+1))/(9^n.3^(n-1))`

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Sequence sums by Archimedean property: lim(n→∞)⁡(3n+1)/(2n+5)=3/2 and lim(n→∞)⁡(n2-1)/(n3+3)=1/2

The value of `i^(1+3+5+...+(2n+1))` is (1) i if n is even, -i if n is odd (2) 1 if n is even, ...

Prove 3^(2n+1) + 2^(n-1) is divisible by 7 using Mathematical Induction