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# Binomial Theorem

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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 + 180x^2 + cx^3 + ...

Find c.

Jul 5, 2022

#1
+2275
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By the Binomial Theorem, we have the system:

$$a \times n = -20$$

$${n \choose 2} \times a^2 = 180$$

Squaring the first equation gives us $$a^2n^2 = 400$$, meaning $${n^2 \over {n \choose 2}} = {20 \over 9}$$. Using trial and error, we find $$n = 10$$, meaning $$a = -2$$

Now, we have $$c = {10 \choose 3} \times 1 \times -2x^3 = \color{brown}\boxed{-960}$$

Jul 5, 2022