- Home
- Standard 11
- Mathematics
If $n$ is even positive integer, then the condition that the greatest term in the expansion of ${(1 + x)^n}$ may have the greatest coefficient also, is
$\frac{n}{{n + 2}} < x < \frac{{n + 2}}{n}$
$\frac{{n + 1}}{n} < x < \frac{n}{{n + 1}}$
$\frac{n}{{n + 4}} < x < \frac{{n + 4}}{4}$
None of these
Solution
(a) If $n$ is even, the greatest coefficient is $^n{C_{n/2}}$
Therefore the greatest term $ = {\,^n}{C_{n/2}}{x^{n/2}}$
$\therefore \,{\,^n}{C_{n/2}}{x^{n/2}} > {\,^n}{C_{(n/2) – 1}}{x^{(n – 2)/2}}$ and
$^n{C_{n/2}}{x^{n/2}} > {\,^n}{C_{(n/2) + 1}}{x^{(n/2) + 1}}$
==> $\frac{{n – \frac{n}{2} + 1}}{{\frac{n}{2}}}x > 1$and $\frac{{\frac{n}{2}}}{{\frac{n}{2} + 1}}x < 1$
==> $x > \frac{{\frac{n}{2}}}{{\frac{n}{2} + 1}}$ and $x < \frac{{\frac{n}{2} + 1}}{{\frac{n}{2}}}$
==> $x > \frac{n}{{n + 2}}$and $x < \frac{{n + 2}}{n}$
