Binomial Theorem – Practice Questions

1.  In the expansion of ${\left(a–3b\right)}^n$ , the sum of 9^th and 10^th term is zero. Find the value of  $\frac{a}{b}$ in terms of  $n$ .

2.  If the coefficient of 4^th, 5^th and 6^th terms in the expansion of  ${\left(1+x\right)}^n$ are in arithmetic sequence, then find the value(s) of $n$ .

3.  If the last term in the expansion of  ${\left(\sqrt{3 }– \frac{1}{\sqrt{3}}\right)}^n$ is $\frac{–{\log }_{2}81}{2^{{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}}$ , find the value of  $n$ .

4.  Find the remainder when  $2^{1003}$ is divided by 7. (Non Calculator question)

5.  If  $x_n–y_n\sqrt{2}={\left(1–\sqrt{2}\right)}^n$  , then show that

       a.  ${x_n}^2–2{y_n}^2={\left(–1\right)}^n$

6. (i) Find the first three terms in the expansion, in ascending powers of $x$ , of ${\left(1–2x\right)}^5$ . [2 marks]

(ii) Given that the coefficient of $x^2$ in the expansion of $\left(1+ax+2x^2\right){\left(1–2x\right)}^5$  is 12, find the value of the constant $a$ . [ 3 marks]

7. a) Use the binomial theorem to expand  ${\left(a+\sqrt{b}\right)}^4$ . 
b) Hence, deduce an expression in terms of  $a$ and  $b$ for ${\left(a+\sqrt{b}\right)}^4+{\left(a–\sqrt{b}\right)}^4$ . 

Q8. Write down and simplify the general term in the binomial expansion of ${\left(2x^2–\frac{d}{x^3}\right)}^7$ , where  $d$ is a constant.
(b) Given that the coefficient of $\frac{1}{x}$ is −70 000, find the value of $d$ .