Let xn be the sequence of numbers denoted by xn=(195/4 Pn)-( n+3 P3/Pn+1)(n ∈ N) where Pn denotes the number of ways in which n distinct things can be arranged on n different places in a definite order. The sum of all possible values of n ∈ N for which xn0, is

Question

Let xn be the sequence of numbers denoted by xn=(195/4 Pn)-( n+3 P3/Pn+1)(n ∈ N) where Pn denotes the number of ways in which n distinct things can be arranged on n different places in a definite order. The sum of all possible values of n ∈ N for which xn>0, is (A) 10 (B) 9 (C) 8 (D) 6

Tardigrade Admin · Accepted Answer

We must have, (2 n +19)(2 n -9)<0 ⇒ (-19/2)< n <(9/2) ⇒ n ∈ 1,2,3,4 Hence, sum =1+2+3+4=10

Q. Let xn​ be the sequence of numbers denoted by xn​=4Pn​195​−Pn+1​n+3P3​​(n∈N) where Pn​ denotes the number of ways in which n distinct things can be arranged on n different places in a definite order. The sum of all possible values of n∈N for which xn​>0, is

10

9

8

6

We must have, (2n+19)(2n−9)<0⇒2−19​<n<29​⇒n∈{1,2,3,4} Hence, sum =1+2+3+4=10

We must have,
$(2 n + 19) (2 n - 9) < 0 \Rightarrow \frac{- 19}{2} < n < \frac{9}{2} \Rightarrow n \in {1, 2, 3, 4}$
Hence, sum $= 1 + 2 + 3 + 4 = 10$