Question -
Answer -
Given:
P(2n – 1, n) : P(2n +1, n – 1) = 22 : 7
P(2n – 1, n) / P(2n +1, n – 1) = 22 / 7
By using the formula,
P (n, r) = n!/(n – r)!
P (2n – 1, n) = (2n –1)! / (2n – 1 – n)!
= (2n – 1)! / (n – 1)!
P (2n + 1, n – 1) =(2n + 1)! / (2n + 1 – n + 1)!
= (2n + 1)! / (n + 2)!
So, from the question,
P(2n – 1, n) / P(2n +1, n – 1) = 22 / 7
Substituting theobtained values in above expression we get,
[(2n– 1)! / (n – 1)!] / [(2n + 1)! / (n + 2)!] = 22/7
[(2n– 1)! / (n – 1)!] × [(n + 2)! / (2n + 1)!] = 22/7
[(2n– 1)! / (n – 1)!] × [(n + 2) (n + 2 – 1) (n + 2 – 2) (n + 2 – 3)!] / [(2n + 1)(2n + 1 – 1) (2n + 1 – 2)] = 22/7
[(2n– 1)! / (n – 1)!] × [(n + 2) (n + 1) n(n – 1)!] / [(2n + 1) 2n (2n – 1)!] =22/7
[(n +2) (n + 1)] / (2n + 1)2 = 22/7
7(n + 2) (n + 1) =22×2 (2n + 1)
7(n2 +n + 2n + 2) = 88n + 44
7(n2 +3n + 2) = 88n + 44
7n2 +21n + 14 = 88n + 44
7n2 +21n – 88n + 14 – 44 = 0
7n2 –67n – 30 = 0
7n2 –70n + 3n – 30 = 0
7n(n – 10) + 3(n – 10)= 0
(n – 10) (7n + 3) = 0
n = 10, -3/7
We know that, n ≠ -3/7
∴ The value of nis 10.