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# Binomial Theorem

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20
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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.

May 2, 2022

#1
+9310
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We need to know a and n first before we can find c.

Note that $$\begin{cases}an = -20\\\dfrac{a^2 n(n - 1)}2 = 180\end{cases}$$. (Use binomial theorem and compare coefficients to get this system of equations.)

Solving gives a = -2 and n = 10.

Now, note that $$c = \dfrac{a^3 n(n-1)(n-2)}{6}$$. Substitute the values of a and n and simplify.

May 2, 2022