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# Binomials

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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 + 10x + 150 x^2 + cx^3 + ...

Find c.

Jan 6, 2023

#1
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$$(1+ax)^n=1+nax+\frac{n(n-1)}{2!}a^2x^2+\frac{n(n-1)(n-2)}{3!}a^3x^3+...$$

Let  $$na=10$$

and    $$\frac{n(n-1)}{2!}a^2=150$$

and solve for a and n, then find c from $$c=\frac{n(n-1)(n-2)}{3!}a^3$$

Jan 6, 2023