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7.Binomial Theorem
hard
The interval in which $x$ must lie so that the greatest term in the expansion of ${(1 + x)^{2n}}$ has the greatest coefficient, is
A
$\left( {\frac{{n - 1}}{n},\frac{n}{{n - 1}}} \right)$
B
$\left( {\frac{n}{{n + 1}},\frac{{n + 1}}{n}} \right)$
C
$\left( {\frac{n}{{n + 2}},\frac{{n + 2}}{n}} \right)$
D
None of these
Solution
(b) Here the greatest coefficient is $^{2n}{C_n}$
$\therefore \,\,{\,^{2n}}{C_n}{x^n}{ > ^{2n}}{C_{n + 1}}{x^{n – 1}} \Rightarrow x > \frac{n}{{n + 1}}$
and $^{2n}{C_n}{x^n} > {\,^{2n}}{C_{n – 1}}{x^{n + 1}} \Rightarrow x < \frac{{n + 1}}{n}$
Hence the required interval is $\left( {\frac{n}{{n + 1}},\,\frac{{n + 1}}{n}} \right)$.
Standard 11
Mathematics
