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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 \[1 - 20x + 150x^2 + cx^3 + \dotsb.\]Find $c$.

 Mar 9, 2023
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The expansion of $(1 + ax)^n$ can be written using the binomial theorem as:

$$(1 + ax)^n = \sum_{k=0}^{n} \binom{n}{k} (ax)^k$$

The first few terms in this expansion are:

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

Simplifying the first three terms:

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

We are given that the first three terms are:

$$1 + 20x + 150x^2 + cx^3 + \cdots$$

Comparing the coefficients of $x^3$ on both sides of the equation, we get:

$$\binom{n}{3}(a)^3 = c$$

To find $c$, we need to find $\binom{n}{3}$ and $a$. We can use the first two terms of the expansion to solve for $a$ and $n$.

From the given first two terms, we have:

$$1 + nax = 1 + 20x$$

This implies that $nax = 20x$ or $a = \frac{20}{n}$.

Substituting this value of $a$ in the expression for the second term:

$$20x = \binom{n}{1} \left(\frac{20}{n}\right)x$$

Simplifying this equation gives:

$$n = 2\binom{n}{1} = 2n$$

Thus, $n = 2$.

Now, we can use $n=2$ and $a=\frac{20}{n}=10$ to find $c$:

$$\binom{2}{3}(10)^3 = 2000$$

Therefore, $c=2000$.

 Mar 9, 2023

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