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# binomial theorem stuff

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For some real number a and some positive integer n, the first few terms in the expansion of (1 + ax)^n are 1 - 20x + 150x^2 + cx^3 + ...

Find c.

Mar 13, 2023

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We know that the first few terms in the expansion of $(1+ax)^n$ are $1-20x+150x^2+cx^3+\cdots$. Let us expand this expression using the binomial theorem:

$$(1+ax)^n = \binom{n}{0}1^n + \binom{n}{1}1^{n-1}(ax) + \binom{n}{2}1^{n-2}(ax)^2 + \binom{n}{3}1^{n-3}(ax)^3 + \cdots$$

Simplifying, we have

$$(1+ax)^n = 1 + anx + \frac{n(n-1)}{2}(ax)^2 + \frac{n(n-1)(n-2)}{6}(ax)^3 + \cdots$$

Comparing this to the values given in the problem, we have

\begin{align*} an &= -20 \ \frac{n(n-1)}{2}a^2 &= 150 \ \frac{n(n-1)(n-2)}{6}a^3 &= c \ \end{align*}

From the first equation, we have $n=-\frac{20}{a}$. Substituting this into the second equation and simplifying, we get $a=-\frac{3}{2}$. Substituting $n$ and $a$ into the third equation and simplifying, we find that $c = \boxed{-\frac{625}{4}}$.

Therefore, $c=-\frac{625}{4}$ is the value we were asked to find.

Mar 13, 2023