The question is this:
In the expansion of $(3 − 2x)(1 + x/2)^n$, the coefficient of $x$ is $7$. Find the value of the constant $n$ and
hence find the coefficient of $x^2$.
If $n$ were given I could have solved it using the binomial theorem but I don't know how to solve problems such as these.
The answer key states this but I cant make any sense of it.
Term in $x = nx/2$
$(3-2x)(1+nx/2+\dots) = 7$
$n=6$
Best Answer
There are only two terms with the first power of $x$, namely $(-2x)(1)$ and $3 \binom n 1 (\frac x 2)$. So $(-2x)(1)+3 \binom n 1 (\frac x 2)=7x$ which gives $n=6$.
When you expand the power using Binomial Theorem and multiply by $3-2x$ you get the first power of $x$ if you multiply the constant term in $3-2x$ by the coefficient of $x$ in $(1+\frac x 2)^{n}$ or you multiply the constant term in $(1+\frac x 2)^{n}$ by the coefficient of $x$ in $3-2x$.