I need to calculate the coefficient of a specific term in a binomial, but how do I do that if the exponent has a variable in it?
For example:
Find the coefficient of $x^n$ in the expansion of $(4x + 5x^2)^{7n}$
or
Find the coefficient of $x^5$ in the expansion of $(3 + 2x^2)^{5n}$
Note that these examples are not a homework problems; I do not need just the answer. I am trying to learn how to solve very similar problems because my text does not explain (and I can't figure it out).
Best Answer
Looking at your first example: $$ (4x + 5x^2)^{7n} = \sum_{k=0}^{7n}\binom{7n}{k}4^kx^k5^{7n-k}x^{2(7n-k)} $$ So, for which of the $k$ we have $k+2(7n-k)=n$? Solving this for $k$ we get $k=13n$ which (for $n>0$) is not included in the sum (so the coefficient of $x^n$ is zero).
The second example is even easier: For any integer $n$, the exponents of $x$ in $(3 + 2x^2)^{5n}$ all are even...
In general, you have to write down the binomial formula (as somthing like $\sum_{k=0}^{f(n)}\ldots$) for your term and solve the desired equation for the exponents of $x$ for $k$. Then, knowing all these $k$, you can evaluate the sum for just these $k$ (which is probably just a single term) with $x=1$ to find the desired coefficient.