If the middle term of (1+x)^2n is the greatest term then X lies between A) n-1 and n B)n/(n+1) - Maths - Mathematical Induction and Binomial Theorem

Question

If the middle term of (1+x)^2n is the greatest term then X lies
between A) n-1 and n B)n/(n+1)

Varun.Rawat · Accepted Answer

The exponent of 1+x in 1+x2n is an even number 2n
So, 2n2+1th i e
 n+1th term is the middle term in the binomial expansion of 1+x2n
Now, Tn+1 = Cn2n 12n-n 
 xn = Cn2n 
 xnNow, we are given that greatest term is Cn2n 
 xn So, Cn-12n 
 xn-1 < Cn2n 
 xn  and   Cn+12n 
 xn+1 < Cn2n 
 xn⇒Cn-12nCn2n < x < Cn2nCn+12n⇒2n!n+1! 
 n-1! × n! 
 n!2n! < x < 2n!n! 
 n! × n+1! 
 n-1!2n!⇒nn+1 < x < n+1n