If 2n+1Pn−1:2n−1Pn=3:5, then n?

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Answer 1 · 2018-08-03T13:59:14

#n=4#

We know that , Permutation

#color(blue)(nP_r=(n!)/((n-r)!) ,where, 1 <=r <= n ,n in N#
We have,

#((2n+1)P_(n-1))/((2n-1)P_n)=3/5#

#=>((2n+1)P_(n-1))/3=((2n-1)P_n)/5#

#=>((2n+1)!)/(((2n+1)-(n-1))!xx3)=((2n-1)!)/(((2n-1)-n)!xx5)#

#=>((2n+1)(2n)color(red)((2n-1))!)/((n+2)!xx3)=((color(red)(2n-1))!)/((n-1)!xx5)#

#=>((2n+1)(2canceln))/((n+2)(n+1)canceln(color(blue)((n-1))!)xx3)=1/(color(blue)((n-1)!)xx5)#

#=>(4n+2)/((n^2+3n+2)xx3)=1/5#

#=>20n+10=3n^2+9n+6#

#=>3n^2-11n-4=0#

#=>3n^2-12n+n-4=0#

#=>3n(n-4)+1(n-4)=0#

#=>(n-4)(3n+1)=0#

#=>n-4=0 or 3n+1=0#

#=>n=4 or n=-1/3 !in N#

So,

#n=4#