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JEE Main 202331 JanuaryEvening Shift
Exam
JEE Main
Year
2023
Date
31 January
Shift
Evening Shift
Subject
Mathematics
Chapter
Binomial Theorem

JEE Main 2023 · 31 January · Evening Shift · Mathematics · Binomial Theorem

If Pn - 1 2n + 1 : Pn 2n - 1 = 11: 21 , then n ^ 2 + n + 15 is...

If Pn−12n+1:Pn2n−1=11:21Pn - 1 2n + 1 : Pn 2n - 1 = 11: 21, then n2+n+15{n}^{2} + n + 15 is equal to :

Correct answer

0

Step-by-step solution

Given,

Pn−12n+1Pn2n−1=1121\dfrac{Pn - 1 2n + 1}{Pn 2n - 1} = \dfrac{11}{21}

⇒(2n+1)!(n+2)!(2n−1)!(n−1)!=1121⇒ \dfrac{\dfrac{\left(2n + 1\right) !}{\left(n + 2\right) !}}{\dfrac{\left(2n - 1\right) !}{\left(n - 1\right) !}} = \dfrac{11}{21}

⇒21×(2n+1)2n(2n−1)!(n+2)(n+1)n(n−1)!=11×(2n−1)!(n−1)!⇒ 21 \times \dfrac{\left(2n + 1\right)2n\left(2n - 1\right) !}{\left(n + 2\right)\left(n + 1\right)n\left(n - 1\right) !} = 11 \times \dfrac{\left(2n - 1\right) !}{\left(n - 1\right) !}

⇒21×2(2n+1)=11(n+2)(n+1)⇒ 21 \times 2\left(2n + 1\right) = 11\left(n + 2\right)\left(n + 1\right)

⇒(2n+1)(n+2)(n+1)=1142⇒ \dfrac{\left(2n + 1\right)}{\left(n + 2\right)\left(n + 1\right)} = \dfrac{11}{42}

Now on comparing both side we get, n=5n = 5

So, the value of n2+n+15=52+5+15=45{n}^{2} + n + 15 = {5}^{2} + 5 + 15 = 45

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