+850 For some real number $a$ and some positive integer $n,$ the first few terms in the expansion of $(1 + ax)^n$ are
\[1 - 6x + 12x^2 + cx^3 + \dotsb.\]
Find $c.$
+399 \(\begin{cases} {n \choose 1}*a = -6 \\ {n \choose 2}*a^2 = 12 \end{cases}\)
\(\begin{cases} na=-6 \\ \frac{n(n-1)}{2}*a*a=12 \end{cases}\)
\(\begin{cases} na=-6 \\ na*a(n-1)=24 \end{cases}\)
\(-6*a(n-1)=24\)
\(a(n-1)=-4\)
\(an-a=-4\)
\(\begin{cases} a = -2\\n=3 \end{cases}\)
\({(1-2x)}^{3}=1-6x+12{x}^{2}-8{x}^{3}\)
\(\bf{c=-8}\).
\(\)
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