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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
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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

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