The binomial theorem：練習問題

In the expansion of \((1 + x)^n\) in ascending powers of \(x\), where \(n\) is a positive integer, the coefficient of the third term is \(45\). Find the value of \(n\).

In the expansion of \((1 + px)^n\) in ascending powers of \(x\), the coefficient of \(x\) is \(32\) and the coefficient of \(x^2\) is \(448\), where \(n\) is a positive integer and \(p\) is a non-zero constant. Find the value of \(n\).

In the expansion of \((1 + ax)^n\) in ascending powers of \(x\), where \(a\) and \(n\) are positive integers, the coefficient of \(x^2\) is \(60\). If \(n = 3a\), find the coefficient of \(x\).

In the expansion of \((1 + 4x)^{12}\), what is the ratio of the coefficient of \(x^2\) to the coefficient of \(x\)?

In the expansion of The

Find the coefficient of \(x^2\) in the expansion of \((1 - 5x)^4\).

In the expansion of \( \left( x^2 - \frac{k}{x} \right)^n \), there are \( 7 \) terms. Given that the coefficient of \( x^3 \) is \( 160 \), find the value of the constant \( k \).

In the expansion of \((1+x)^{10}\), let \(C_r\) denote the coefficient of \(x^r\). Given that the ratio \(C_{r+1}:C_r\) is \(3:8\), find the value of \(r\).

(a) Write down the first three terms, in ascending powers of \(x\), in the expansion of \((1+2x)^n\).
(b) Given that the coefficient of \(x^2\) in the expansion of \((1+2x)^n\) is \(120\), find the value of the positive integer \(n\).

In the expansion of $$(1+ax)^5(1-2x)^6$$, the coefficient of $$x$$ is $$-17$$.
(a) Find the value of the constant $$a$$.
(b) Using the value of $$a$$ found in (a), find the coefficient of $$x^2$$ in the expansion of $$(1+ax)^5(1-2x)^6$$.