# University Maths Solution

Maths Question
Question 1

$\text{Prove by mathematical induction or otherwise that }\sum\nolimits_{r=1}^{n}{r}=\tfrac{n(n+1)}{2}$

Question 2

$\text{Prove by mathematical induction or otherwise that }\sum\nolimits_{r=1}^{n}{{{r}^{2}}}=\tfrac{n}{6}(n+1)(2n+1)$

Question 3

$\text{Prove by mathematical induction or otherwise that }\sum\nolimits_{r=1}^{n}{{{r}^{3}}}={{\left[ \tfrac{1}{2}n(n+1) \right]}^{2}}$

Question 4

\begin{align} & \text{Use the method of mathematical induction to establish} \\ & 1+3+5+\cdot \cdot \cdot +(2n-1)={{n}^{2}} \\ & n\text{ is assumed to be positive integer} \\\end{align}

Question 5

\begin{align} & \text{Use the method of mathematical induction to establish } \\ & n(n+1)(n+2)\text{ is an integer multiple of 6} \\ & n\text{ is assumed to be positive integer} \\\end{align}

Question 6

\begin{align} & \text{Use the method of mathematical induction to establish} \\ & (1\times 2\times 3)+(2\times 3\times 4)+\cdot \cdot \cdot +n(n+1)(n+2)=\frac{n(n+1)(n+2)(n+3)}{4} \\ & n\text{ is assumed to be positive integer} \\\end{align}

Question 7

\begin{align} & \text{Use the method of mathematical induction to establish} \\ & \text{ }{{2}^{n}}>n \\ & n\text{ is assumed to be positive integer} \\\end{align}

Question 8

\begin{align} & \text{Use the method of mathematical induction to establish } \\ & {{7}^{2n+1}}\text{+1 is divisible by 8} \\ & n\text{ is assumed to be positive integer} \\\end{align}

Question 9

Use mathematical induction to prove the given formula for every positive integer n

$\sum\limits_{r=1}^{n}{(r+1)\cdot {{2}^{r}}}=n\cdot {{2}^{n+1}}$

Question 10

Use the principle of mathematical induction to show that 17 is a factor of ${{3}^{4n+2}}+2\cdot {{4}^{3n+1}}$

Question 11

Use the principle of mathematical induction to show that 7 is a factor of ${{4}^{3n-1}}+{{2}^{3n-1}}+1$