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If \( { }^{2 n+1} P_{n-1}:{ }^{2 n-1} P_{n}=3: 5 \), find the value of \( n \).

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If \( { }^{2 n+1} P_{n-1}:{ }^{2 n-1} P_{n}=3: 5 \), find the value of \( n \).
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Solving we get

⇒20n+10=3n2+9n+6

⇒3n2−11n−4=0

⇒3n2−12n+n−4=0

⇒3n(n−4)+1(n−4)=0

∴n=4

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