Question

If the middle term of (1+ x)²ⁿ (x > 0, n Ī N) is the greatest term of the expansion. Then the interval in which n lies, is-

• A. $\left\lbrack \frac{n + 1}{n},\frac{n + 2}{n} \right\rbrack$
• B. $\left\lbrack \frac{n - 1}{n},\frac{n + 1}{n} \right\rbrack$
• C. $\left\lbrack \frac{n}{n + 1},\frac{n + 1}{n} \right\rbrack$
• D. None of these

Answer 1

Answer: (C)

$\left\lbrack \frac{n}{n + 1},\frac{n + 1}{n} \right\rbrack$

Answer 2

Middle term = ²ⁿC_n .xⁿ = t_n+1

tⁿ⁺¹ is also greatest – therefore

t_n+1 > t_n …(1)

t_n+1 > t_n+2 …(2)

From (1)

Ž ²ⁿC_n. xⁿ > ²ⁿC_n–1. x^n–1 Ž $\frac{2n!}{n!.n!}$. x >$\frac{2n!}{n - 1!.n + 1!}$

Ž (n + 1) x > n Ž x > $\frac{n}{n + 1}$ … (i)

From (2)

Ž ²ⁿC_n. xⁿ > ²ⁿC_n+1. xⁿ⁺¹

Ž $\frac{2n!}{n!.n!}$> $\frac{2n!}{n - 1!.n + 1!}$x Ž x < $\frac{n + 1}{n}$ … (ii)

So, from (i) and (ii), x Ī $\left( \frac{n}{n + 1},\frac{n + 1}{n} \right)$

Now, we know that the greatest term can also be equal to one of the adjacent terms. Hence the equality can also holds with equation (1) or equation (2)

\ x Ī $\left\lbrack \frac{n}{n + 1},\frac{n + 1}{n} \right\rbrack$