Question

if p and q are the coefficients of x power n in the expansion of oneplus x whole power 2n and one plus x whole power 2 and -1 respectively then p by q is equals to

Answer 1

Given :   p and q are the coefficients of xⁿ in the expansion of (1 + x)²ⁿ  and (1 + x)²ⁿ⁻¹ respectively

To Find : p/q

Solution:

coefficients of xⁿ in the expansion of (1 + x)²ⁿ  

= ²ⁿCₙ

=> p = ²ⁿCₙ

=> p = (2n)!/n!.(2n-n)!

=> p = (2n)!/n!.n!

coefficients of xⁿ in the expansion of (1 + x)²ⁿ⁻¹  

= ²ⁿ⁻¹Cₙ

=> q = ²ⁿ⁻¹Cₙ

=> q = (2n-1)!/n!.(2n-1-n)!

=> q = (2n)!/n!.(n-1)!

p = (2n)!/n!.n!

=> p = 2n.(2n-1)!/n!. n(n-1)!

=> p = 2 . (2n)!/n!.(n-1)!

=> p = 2.q

=> p/q  = 2

Hence p/q = 2

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