### Binomial Theorem

#### Expand binomials with positive integral indices using the Binomial Theorem

• It is given that $${(1 + 4x)^n} - {(1 - 3x)^n} = 7nx + 196{x^2} +$$ terms involving higher powers of x.
(a) If n is a positive integer, find the value of n.
(b) Hence find the coefficient of $${x^3}$$.
• (a) $${(1 + 4x)^n} = 1 + C_1^n(4x) + C_2^n{(4x)^2} +$$ terms involving higher powers of x
$$= 1 + 4nx + 8n(n - 1){x^2} +$$ terms involving higher powers of x
$${(1 - 3x)^n} = 1 + C_1^n( - 3x) + C_2^n{( - 3x)^2} +$$ terms involving higher powers of x
$$= 1 - 3nx + \frac{{9n(n - 1)}}{2}{x^2} +$$ terms involving higher powers of x
∴ $${(1 + 4x)^n} - {(1 - 3x)^n} = [4n - ( - 3n)]x + [8n(n - 1) - \frac{{9n(n - 1)}}{2}]{x^2}$$ + terms involving higher powers of x                                              $$= 7nx + \frac{{7n(n - 1)}}{2}{x^2} +$$ terms involving higher powers of x
By comparing the coefficient of $${x^2}$$, we have
$$\begin{array}{1}\frac{{7n(n - 1)}}{2} = 196\\n(n - 1) = 56\\{n^2} - n - 56 = 0\\(n - 8)(n + 7) = 0\end{array}$$
$$n = \underline{\underline 8}$$ or $$n = - 7$$ (rejected)