Question

The sum of the coefficients in the first three terms of the expansion of ${(x^2-\frac{2}{x})}^m$ is equal to $97$. Find the term of the expansion containing $x^4$.

Answer 1

Sum of first three terms be ${}^m{C}_0-2^m{C}_1+4^m{C}_2=97$

$⟹1-2m+2\left(m\right)(m-1)=97⟹2m^2-4m-96=0⟹m^2-2m-48=0⟹m=8,-6$

$m$ can't be negative so $m=8$

General term is ${}^8{C}_r\left(x^{16-2r}\right)(\frac{-2}{x}{)}^r$

$16-3r=4⟹r=4$

The term of expansion containing $x^4$ is ${}^8{C}_4(2{)}^4{x}^4$