Prove, by mathematical induction, that \(\sum_{r=1}^{n} r \ln \left(\frac{r+1}{r}\right)=\ln \left(\frac{(n+1)^{n}}{n!}\right)\) for all positive integers \(n\).
Given that \(a\) is a constant, prove by mathematical induction that, for every positive integer \(n\),
\(\frac{\mathrm{d}^{n}}{\mathrm{~d} x^{n}}\left(x \mathrm{e}^{a x}\right)=n a^{n-1} \mathrm{e}^{a x}+a^{n} x \mathrm{e}^{a x} .\)
Prove by mathematical induction that, for all positive integers \(n, \sum_{r=1}^{n} \frac{1}{(2 r)^{2}-1}=\frac{n}{2 n+1}\).
State the value of \(\sum_{r=1}^{\infty} \frac{1}{(2 r)^{2}-1}\).
The sequence \(a_{1}, a_{2}, a_{3}, \ldots\) is such that \(a_{1}\gt 5\) and \(a_{n+1}=\frac{4 a_{n}}{5}+\frac{5}{a_{n}}\) for every positive integer \(n\). Prove by mathematical induction that \(a_{n}\gt 5\) for every positive integer \(n\).
Prove also that \(a_{n}\gt a_{n+1}\) for every positive integer \(n\).
Using factorials, show that \(\binom{n}{r-1}+\binom{n}{r}=\binom{n+1}{r}\).
Hence prove by mathematical induction that
\((a+x)^{n}=\binom{n}{0} a^{n}+\binom{n}{1} a^{n-1} x+\ldots+\binom{n}{r} a^{n-r} x^{r}+\ldots+\binom{n}{n} x^{n}\)
for every positive integer \(n\).
It is given that a diagonal of a polygon is a line joining two non-adjacent vertices. Prove, by mathematical induction, that an \(n\)-sided polygon has \(\frac{1}{2} n(n-3)\) diagonals, where \(n \geqslant 3\).
(i) Show that \(\frac{\mathrm{d}^{n+1}}{\mathrm{~d} x^{n+1}}\left(x^{n+1} \ln x\right)=\frac{\mathrm{d}^{n}}{\mathrm{~d} x^{n}}\left(x^{n}+(n+1) x^{n} \ln x\right)\).
(ii) Prove by mathematical induction that, for all positive integers \(n\),
\(\frac{\mathrm{d}^{n}}{\mathrm{~d} x^{n}}\left(x^{n} \ln x\right)=n!\left(\ln x+1+\frac{1}{2}+\ldots+\frac{1}{n}\right)\)
It is given that \(u_{r}=r \times r!\) for \(r=1,2,3, \ldots\). Let \(S_{n}=u_{1}+u_{2}+u_{3}+\ldots+u_{n}\). Write down the values of
\(2!-S_{1}, \quad 3!-S_{2}, \quad 4!-S_{3}, \quad 5!-S_{4} .\)
Conjecture a formula for \(S_{n}\).
Prove, by mathematical induction, a formula for \(S_{n}\), for all positive integers \(n\).
It is given that \(u_{r}=r \times r!\) for \(r=1,2,3, \ldots\). Let \(S_{n}=u_{1}+u_{2}+u_{3}+\ldots+u_{n}\). Write down the values of
\(2!-S_{1}, \quad 3!-S_{2}, \quad 4!-S_{3}, \quad 5!-S_{4} .\)
Conjecture a formula for \(S_{n}\).
Prove, by mathematical induction, a formula for \(S_{n}\), for all positive integers \(n\).
Prove by mathematical induction that, for every positive integer \(n\),
\(\frac{\mathrm{d}^{n}}{\mathrm{~d} x^{n}}\left(\mathrm{e}^{x} \sin x\right)=(\sqrt{2})^{n} \mathrm{e}^{x} \sin \left(x+\frac{1}{4} n \pi\right) .\)
It is given that \(y=(1+x)^{2} \ln (1+x)\). Find \(\frac{\mathrm{d}^{3} y}{\mathrm{~d} x^{3}}\).
Prove by mathematical induction that, for every integer \(n \geqslant 3\),
\(\frac{\mathrm{d}^{n} y}{\mathrm{~d} x^{n}}=(-1)^{n-1} \frac{2(n-3)!}{(1+x)^{n-2}}\)
It is given that \(y=(1+x)^{2} \ln (1+x)\). Find \(\frac{\mathrm{d}^{3} y}{\mathrm{~d} x^{3}}\).
Prove by mathematical induction that, for every integer \(n \geqslant 3\),
\(\frac{\mathrm{d}^{n} y}{\mathrm{~d} x^{n}}=(-1)^{n-1} \frac{2(n-3)!}{(1+x)^{n-2}}\)
Prove by mathematical induction that, for every positive integer \(n\),
\((\cos \theta+\mathrm{i} \sin \theta)^{n}=\cos n \theta+\mathrm{i} \sin n \theta .\)
Express \(\sin ^{5} \theta\) in the form \(p \sin 5 \theta+q \sin 3 \theta+r \sin \theta\), where \(p, q\) and \(r\) are rational numbers to be determined.
Prove by induction that
\(\sum_{r=1}^{n}\left(3 r^{5}+r^{3}\right)=\frac{1}{2} n^{3}(n+1)^{3}\)
for all \(n \geqslant 1\).
Use this result together with the List of Formulae (MF10) to prove that
\(\sum_{r=1}^{n} r^{5}=\frac{1}{12} n^{2}(n+1)^{2} \mathrm{Q}(n),\)
where \(\mathrm{Q}(n)\) is a quadratic function of \(n\) which is to be determined.
For the sequence \(u_{1}, u_{2}, u_{3}, \ldots\), it is given that \(u_{1}=1\) and \(u_{r+1}=\frac{3 u_{r}-2}{4}\) for all \(r\). Prove by mathematical induction that \(u_{n}=4\left(\frac{3}{4}\right)^{n}-2\), for all positive integers \(n\).
Let \(S_{N}=\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\ldots+\frac{N}{(N+1)!}\). Prove by mathematical induction that, for all positive integers \(N\),
\(S_{N}=1-\frac{1}{(N+1)!} .\)
Let \(\mathbf{A}=\left(\begin{array}{ll}2 & 3 \\ 0 & 1\end{array}\right)\). Prove by mathematical induction that, for every positive integer \(n\),
\(\mathbf{A}^{n}=\left(\begin{array}{cc} 2^{n} & 3\left(2^{n}-1\right) \\ 0 & 1 \end{array}\right) .\)
Prove by mathematical induction that, for all positive integers \(n\),
\(\frac{\mathrm{d}^{n}}{\mathrm{~d} x^{n}}\left(\mathrm{e}^{x} \sin x\right)=2^{\frac{1}{2} n} \mathrm{e}^{x} \sin \left(x+\frac{1}{4} n \pi\right) .\)
Prove by mathematical induction that, for all positive integers \(n\),
\(\frac{\mathrm{d}^{n}}{\mathrm{~d} x^{n}}\left(\frac{1}{2 x+3}\right)=(-1)^{n} \frac{n!2^{n}}{(2 x+3)^{n+1}}\)
The sequence \(x_{1}, x_{2}, x_{3}, \ldots\) is such that \(x_{1}=3\) and
\(x_{n+1}=\frac{2 x_{n}^{2}+4 x_{n}-2}{2 x_{n}+3}\)
for \(n=1,2,3, \ldots\). Prove by induction that \(x_{n}\gt 2\) for all \(n\).