Binomial theorem with fractional/negative indices: notes and practice questions

A hotel has five vacant rooms in a row, labelled 1 to 5. Four new guests, Alice, Ben, Chloe, and David, are to be assigned to these rooms. Alice and Ben are a couple who have recently had an argument and wish to be in separate rooms.

(a) The rooms are large suites, and each suite can accommodate all four guests. Find the number of ways the guests can be assigned to the rooms if Alice and Ben must be in different rooms.

(b) Each room can only accommodate one guest. Find the number of ways the guests can be assigned to the rooms if Alice and Ben must not be in adjacent rooms.

Consider the expression $\sqrt{\frac{1+ax}{1-2x}}$ where $a \in \mathbb{Z}$.

The binomial expansion of this expression, in ascending powers of $x$, up to and including the term in $x^2$ is $1 + 3x + bx^2$, where $b \in \mathbb{Q}$.

(a) Find the value of $a$ and the value of $b$.

(b) Hence, state the interval of convergence for the expansion.

Consider the expression $f(x) = \frac{1}{1+4x}$, where $|x| < \frac{1}{4}$.

(a) (i) Show that the first four terms in the binomial expansion of $f(x)$ are $1 - 4x + 16x^2 - 64x^3$.

(ii) Hence, find an approximation for $\int \frac{1}{1+4x} dx$.

(b) (i) Use your result from part (a)(ii) to find an approximate value for $\int_0^{1/8} \frac{1}{1+4x} dx$.

(ii) Hence, find an approximation for $\ln(1.5)$ in the form $\frac{p}{q}$ where $p, q \in \mathbb{Z}$.

Let $f(x) = \frac{3x^2 + 5x + 1}{(1+x)^2(2-x)}$.

(a) Express $f(x)$ in the form $\frac{A}{1+x} + \frac{B}{(1+x)^2} + \frac{C}{2-x}$.

(b) Hence, find the Maclaurin series for $f(x)$ up to and including the term in $x^3$.

(c) State the interval of convergence for this series.

The function $f$ is defined by $f(x) = (1-2x)^{-3}$. The first three terms of the Maclaurin series for $f(x)$ are $1 + ax + bx^2 + \dots$.

(a) (i) Show that $a=6$.

(ii) Find the value of $b$.

(b) The value of a rare collectible is modelled by the function $V(t) = 1000(1-0.05t)^{-3}$ dollars, where $t$ is the number of years from the start of 2024. Use the first three terms of the Maclaurin series for $f(x)$ to estimate the value of the collectible at the start of 2026.

Consider the expression $\sqrt{1+ax} - (1-x)^{-2}$ where $a \in \mathbb{Z}, a \neq 0$.

The binomial expansion of this expression, in ascending powers of $x$, up to and including the term in $x^2$, is $kx + \frac{3}{2}kx^2$, where $k \in \mathbb{Z}$.

Find the value of $a$ and the value of $k$.

State the restriction which must be placed on $x$ for this expansion to be valid.

Write down the first three terms of the binomial expansion of $(1-t)^{-2}$ in ascending powers of $t$.

The Maclaurin series for $\cos x$ is given by $\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots$

By using this series and the result from part (a), show that the Maclaurin series for $\sec^2 x$ up to and including the term in $x^4$ is $1 + x^2 + \frac{2x^4}{3}$.

The Maclaurin series for $\sin x$ is given by $\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots$

By using this series and the result from part (b), find

$\lim_{x \to 0} \frac{x \sin 3x}{\sec^2 x - 1}$.

The coefficient of $x^2$ in the binomial expansion of $(1-4x)^n$ is 720, where $n \in \mathbb{Q}$.

(a) Determine the possible values of $n$.

(b) For the negative value of $n$ found in part (a), find the interval of convergence for the expansion.

A civil engineer is calculating the stress distribution in a new material. Part of their calculation involves expanding the expression $(1+x)^{1/3}$ for small values of $x$.

(a) Find the first four terms, in ascending powers of $x$, of the binomial expansion of $(1+x)^{1/3}$, stating the condition for which the expansion is valid.

To verify some experimental data, the engineer needs to approximate the value of $\sqrt[3]{28}$.

(b) Use your answer to part (a) to find an approximation for $\sqrt[3]{28}$ to six decimal places. You must show all your working.

Consider the function $f(x) = \frac{3x+5}{1+2x-3x^2}$.

(a) Express $f(x)$ in partial fractions.

(b) Hence, find the binomial expansion of $f(x)$ in ascending powers of $x$, up to and including the term in $x^3$.

(c) State the interval of convergence for this expansion.

(a) Write down and simplify the first three terms, in ascending powers of $x$, in the Extended Binomial expansion of $(1 + x)^{\frac{1}{2}}$ .

(b) By substituting a suitable value of $x$, find a rational approximation to $\sqrt{10}$.

Consider the identity $\frac{5x-5}{(2x+1)(x-2)} \equiv \frac{P}{2x+1} + \frac{Q}{x-2}$, where $P, Q \in \mathbb{Z}$.

Find the value of $P$ and the value of $Q$.

Hence, find the Maclaurin series for $\frac{5x-5}{(2x+1)(x-2)}$ in ascending powers of $x$, up to and including the term in $x^2$.

Explain why the Maclaurin series found in part (b) is not valid for $x=0.6$.