Proof by Induction (Edexcel International A Level (IAL) Further Maths: Further Pure 1): Exam Questions

Prove by induction that for $n\in {\mathrm{\mathbb{Z}}}^{+}$

${\left(\begin{array}{cc}1& r\\ 0& 2\end{array}\right)}^n=\left(\begin{array}{cc}1& (2^n-1)r\\ 0& 2^n\end{array}\right)$

where $r$ is a constant.

$\mathbf{M}=\left(\begin{array}{cc}4& 0\\ 0& 5\end{array}\right)$ $\mathbf{N}={\left(\begin{array}{cc}1& -2\\ 0& 2\end{array}\right)}^4$

The transformation represented by matrix M followed by the transformation represented by matrix N is represented by the matrix B

(i) Determine N in the form $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ where $a$, $b$, $c$ and $d$ are integers.

[1]

(ii) Determine B

[2]

Hexagon $S$ is transformed onto hexagon $S\text{'}$ by matrix B

Given that the area of $S\text{'}$ is 720 square units, determine the area of $S$

Prove by induction that for $n\in {\mathrm{\mathbb{Z}}}^{+}$

$\mathrm{f}\left(n\right)=7^{n-1}+8^{2n+1}$

is divisible by 57

Prove by induction that for $n\in {\mathrm{\mathbb{Z}}}^{+}$

${\left(\begin{array}{cc}5& -1\\ 4& 1\end{array}\right)}^n=3^{n-1}\left(\begin{array}{cc}2n+3& -n\\ 4n& 3-2n\end{array}\right)$

Prove by induction that for $n\in {\mathrm{\mathbb{Z}}}^{+}$

$\mathrm{f}\left(n\right)=8^{2n+1}+6^{2n-1}$

is divisible by 7

Prove, by induction, that for $n\in \mathrm{\mathbb{Z}}$, $n⩾2$

$4^n+6n-10$

is divisible by 18

Prove by induction that for all positive integers $n$

$\sum _{r=1}^n\log (2r-1)=\log \left(\frac{\left(2n\right)!}{2^n n!}\right)$

A sequence of numbers is defined by

$u_1=3$

$u_{n+1}=2u_n-2^{n+1}n⩾1$

Prove by induction that, for $n\in \mathrm{\mathbb{N}}$

$u_n=5\times 2^{n-1}-n\times 2^n$

Prove by induction that, for $n\in \mathrm{\mathbb{N}}$

$\mathrm{f}\left(n\right)=5^{n+2}-4n-9$

is divisible by 16

Prove by induction that, for $n\in \mathrm{\mathbb{N}}$

$\sum _{r=1}^n{r}^3=\frac{1}{4}{n}^2(n+1{)}^2$

Using the standard summation formulae, show that

$\sum _{r=1}^{n}r(r+1)(r-1)=\frac{1}{4}n(n+A)(n+B)(n+C)$

where $A$, $B$ and $C$ are constants to be determined.

Determine the value of $n$ for which

$3\sum _{r=1}^{n}r(r+1)(r-1)=17\sum _{r=n}^{2n}{r}^2$

A sequence of numbers is defined by

$u_1=0u_2=-6$

$u_{n+2}=5u_{n+1}-6u_{n}n\ge 1$

Prove by induction that, for $n\in {\mathrm{\mathbb{Z}}}^{+}$

$u_n=3\times 2^n-2\times 3^n$

Prove by induction that, for all positive integers $n$,

$f\left(n\right)=3^{3n-2}+2^{4n-1}$

is divisible by 11

Prove by induction that for $n\in \mathrm{\mathbb{N}}$

$\sum _{r=1}^n{r}^2=\frac{n}{6}(n+1)(2n+1)$

Hence show that

$\sum _{r=1}^n(r^2+2)=\frac{n}{6}(a{n}^2+bn+c)$

where $a$, $b$ and $c$ are integers to be found.

Using your answers to part (b), find the value of

$\sum _{r=10}^{25}(r^2+2)$

Prove by induction that $4^{n+2}+5^{2n+1}$ is divisible by 21 for all positive integers $n$.

A sequence of numbers $u_1$, $u_2$, $u_3$,....is defined by

$u_{n+1}=\frac{1}{3}(2u_n-1),u_1=1$

Prove by induction that, for $n\in {\mathrm{\mathbb{Z}}}^{+}$

$u_n=3{\left(\frac{2}{3}\right)}^n-1$

$f\left(n\right)=2^{n+2}+3^{2n+1}$

Prove by induction that, for $n\in {\mathrm{\mathbb{Z}}}^{+}$ , $f\left(n\right)$ is a multiple of 7.

(i) Prove by induction that, for $n\in {\mathrm{\mathbb{Z}}}^{+}$

$\sum _{r=1}^n\frac{2r^2-1}{r^2(r+1{)}^2}=\frac{n^2}{(n+1{)}^2}$

[6]

(ii) Prove by induction that, for $n\in {\mathrm{\mathbb{Z}}}^{+}$

$\mathrm{f}\left(n\right)={12}^n+2\times 5^{n-1}$

is divisible by 7.

[6]

Prove by induction that, for $n\in {\mathrm{\mathbb{Z}}}^{+}$

$\mathrm{f}\left(n\right)={12}^n+2\times 5^{n-1}$

is divisible by 7.