$.$ Prove that tFT$2$ tFT$1(1+$ T$1$rT$2)+$ T$1$CT$2$ using a pure arbitrage argument

where t is today, T$1$ and T$2$ are maturity dates and t $<$ T$1<$ T$2$

Ignore marking$-$to$-$the$-$market.

$2.$ a$)$ What are some reasons that in reality this is not a pure arbitrage?

b$)$ How important an effect might these have on the arbitrage profit?

c$)$ Can you think of any way to get around these problems?$$

$3.$ Suppose a trader observes the following prices on June $15$: $(6/15=$ t$)$

Futures price of gold for September $15$ delivery $($T$1=9/15)$ is $$450/$oz$.$

Futures price of gold for December $15$ delivery $($T$2=12/15)$ is $$460/$oz$.$

The borrowing and lending rate is $6\%$ per annum and T$1$CT$2$ is $$2\mathrm{.}00$ payable on $12/15.$

What is the arbitrage profit?

$4.$ Assume that tFT$2=$ tFT$1(1+$ T$1$rT$2)+$ T$1$CT$2.$

Also assume that T$2-$ T$1$ is one year, T$1$CT$2=$ $$1$ and T$1$rT$2=$l$0\%$

A$)$ Are the following equilibrium prices? tFT$1=100,$ tFT$2=133.$

If not, what is the arbitrage profit $($established at time t$)$ from positions held till maturity?

B$)$ Say prices adjust the next day t $+1($ where t $+$ l $<$ T$1)$ where t$+1$FT$1$ can assume the following

possible values:

a$)$ t$+1$FT$1=110$

b$)$ t$+1$FT$1=100$

c$)$ t$+1$FT$1=120$

i$)$ What would be the equilibrium prices for t$+1$FT$2$ in each of these $3$ cases?

ii$)$ What are the resulting profits obtained if you round trip the positions taken in A$)$ for

each of these $3$ equilibrium cases?

iii$)$ What happened to the arbitrage profit $`$locked$-$in$$ in part i$)?$

Is it lost since the market is now in equilibrium or can it be $$regained$?$

iv$)$ Say on day t$+$l$,$ the following prices were observed: t$+1$FT$1=90,$ t$+1$FT$2=135.$

What would you do$?$

Can you explain step by step with calculations so I can better understand these concepts and study from this?