What is the solution to this equation: (2n+1)!/(2n-1)! =42 ;

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thilina-g | College Teacher | (Level 1) Educator

Posted on

You want to solve,

`((2n+1)!)/((2n-1)!) = 42`

I can expand (2n+1)! as below.

`((2n+1)xx2nxx(2n-1)!)/((2n-1)!) = 42`

This gives,

`(2n+1)xx2n = 42`

`(2n+1) xx n = 21`

`2n^2+n-21 = 0`

The solutions for above equation are,

`n = (-1+-sqrt(1+4xx2xx21))/(2xx2)`

`n = (-1+-sqrt(169))/4`

`n = (-1+-13)/4`

`n = 12/4 or n = -14/4`

We know `n!= -14/4` since factorial is not defined for negative numbers.

 

Therefore,

`n = 12/4 = 3`

 

The answer is 3.

 

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