Question $2$

$($a$)$ Use mathematical induction to prove whether that for any positive number $n,n^3+2n$ is divisible by $3$ or not. Show your work

$4$

$($b$)$ Suppose that a flu is spreading through a population. Initially, only one person is infected with the flu. It is also known that an infected person will infect two persons after one round of infection $($and will not infect more people$).$Therefore, after first round of infection, there are two newly infected person each of which will infect two persons further and this continues unbounded.Your task is to derive the mathematical form of the rule that tells:

i$.$ Newly infected persons after $n$ rounds of infection, state if this is an arithmetic or a geometric sequence, what is the value of the difference $d$ if it is an arithmetic sequence or the value of the ratio $r$ if it is a geometric sequence.

ii$.$ The total number of infected persons after $n$ rounds of infection

iii. If we let the disease spread at the same rate, how many persons will be infected after $5$ rounds?

Show your work in each of the above

$[9]$

$($c$)$ Prove that the series whose $n^{th}$ term is $\frac{1}{n(n+1)}$ converges, and find its sum. Show your work

$5$

$($d$)$ Find the sum of all positive integers, from $5$ to $1555$ inclusive, that are divisible by $5.$ Show your work.

$[4]$

$($e$)$ The first and eighth terms of an arithmetic series are $51$ and $100$ respectively.

a$)$ Find the twentieth term of the series. Show your work.

b$)$ Determine the sum of the first twenty terms of the series. Show your work.

$[8]$