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.

Guest May 2, 2022

#1**+1 **

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.

MaxWong May 2, 2022