Question

Find $a,b$ and $n$ in the expansion of $(a+b{)}^n$ if the first three terms of the expansion are $729,7290$ and $30375$, respectively. then $abn$ is equal to

Answer 1

${(a+b)}^n=n_{c_0}{a}^n{b}^0+n_{c_1}{a}^{n-1}b+n_{c_2}{a}^{n-2}{b}^2+..........$

Given

$n_{c_0}{a}^n=729,n_{c_1}{a}^{n-1}b=729$

& $n_{c_2}.a^{n-2}{b}^2=30375$

$a^n=3^6$

$n=6$ ,$a=3$

$6_{c_1}{.3}^5.b=7290$

$6\times 3^5.b=7290$

$b=\frac{1215}{3^5}$

$b=5$

Hence, $n=6,a=3,b=5$

$\therefore abn=90$