Question

If $g\left(x\right)={\sum }_{r=0}^{200}{\alpha }_r{x}^r$ and $f\left(x\right)={\sum }_{r=0}^{200}{\beta }_r{x}^r,{\beta }_r=1$ for $r\ge 100$ and $g\left(x\right)=f(1+x)$, then the greatest coefficient in these expansion of ${(1+x)}^{201}$ is

• A. ${\alpha }_{99}$
• B. ${\alpha }_{100}$
• C. ${\alpha }_{90}$
• D. ${\alpha }_{101}$

Answer 1

The correct option is B ${\alpha }_{100}$

There will be $202$ terms in the expansion of $(1+x{)}^{201}$.
Hence the greatest coefficient will be the coefficient of the middle term.
Now there will be $2$ middle terms.
$T_{101}$ and $T_{102}$
Hence the greatest coefficient is
$={}^{201}{C}_{100}$
$={\alpha }_{100}$.