If P n - 1 2 n + 1 : P n 2 n - 1 = 3 : 5 , then the value of n is equal to

Question

If ${}^{2 n + 1}P_{n - 1} : {}^{2 n - 1}P_{n} = 3 : 5$ , then the value of $n$ is equal to

Smart Achievers · Accepted Answer

(1)

We have

$\frac{{}^{2 n + 1}P_{n - 1}}{{}^{2 n - 1}P_{n}} = \frac{3}{5}$ $\Rightarrow \frac{(2 n + 1)!}{(n + 2)!} \times \frac{(n - 1)!}{(2 n - 1)!} = \frac{3}{5}$ $\Rightarrow \frac{2 (2 n + 1)}{(n + 2) (n + 1)} = \frac{3}{5}$

$\Rightarrow 10 (2 n + 1) = 3 (n + 2) (n + 1) \Rightarrow 3 n^{2} - 11 n - 4 = 0$

$\Rightarrow (3 n + 1) (n - 4) = 0$

$∴$ $n = 4$ $[∵ n \neq \frac{- 1}{3}]$

Solution

Q.
If ${}^{2 n + 1}P_{n - 1} : {}^{2 n - 1}P_{n} = 3 : 5$ , then the value of $n$ is equal to

1 4

2 3

3 2

4 1

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Solution

Q. If Pn-12n+1:Pn2n-1=3:5, then the value of n is equal to

1 4

2 3

3 2

4 1

Ans. (1) We have P n-1 2n+1P n 2n-1=35⇒(2n+1)!(n+2)!×(n-1)!(2n-1)!=35⇒2(2n+1)(n+2)(n+1)=35 ⇒10(2n+1)=3(n+2)(n+1)⇒3n2-11n-4=0 ⇒(3n+1)(n-4)=0 ∴ n=4 [∵ n≠-13]

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Q.
If ${}^{2 n + 1}P_{n - 1} : {}^{2 n - 1}P_{n} = 3 : 5$ , then the value of $n$ is equal to