+0  
 
0
6
1
avatar+383 

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 $c.$

 Mar 21, 2024
 #1
avatar+394 
+1

\(\begin{cases} {n \choose 1}*a = -6 \\ {n \choose 2}*a^2 = 12 \end{cases}\)

\(\begin{cases} na=-6 \\ \frac{n(n-1)}{2}*a*a=12 \end{cases}\)

\(\begin{cases} na=-6 \\ na*a(n-1)=24 \end{cases}\)

\(-6*a(n-1)=24\)

\(a(n-1)=-4\)

\(an-a=-4\)

\(\begin{cases} a = -2\\n=3 \end{cases}\)

\({(1-2x)}^{3}=1-6x+12{x}^{2}-8{x}^{3}\)

\(\bf{c=-8}\).

\(\)

.
 Mar 21, 2024

0 Online Users