Consider the finite$-$horizon Nash bargaining problem discussed in section $12\mathrm{.}9$ of the textbook. Keep everything

the same, except that the number of periods is changed from $2$ to $2$n$.$ We call two consecutive periods a

super$-$period. For example, period $1$ and period $2$ are called super$-$period $1.$ Similarly, periods $3$ and $4$ are

called super$-$period $2.$ In general, super$-$period i$,$ where i $=1,2,...,$ n$,$ contains two periods. The first period

is period $2$i$1$ and the second period is period $2$i$.$ Node A proposes a split in the first period of super$-$periods

and Node B proposes a split in the second period of super$-$periods. Denote the split proposed by node A in

super$-$period i by $($a$1($i$),$ b$1($i$)),$ i $=1,2,...,$ n$.$ Similarly, denote the split proposed by node B in super$-$period

i by $($a$2($i$),$ b$2($i$)).$

a$)$ Derive $($a$1($i$),$ b$1($i$))$ and $($a$2($i$),$ b$2($i$))$ for i $=1,2,...,$ n$.$

b$)$ Show that as n goes to infinity, the limits of the splits derived in part $($a$)$ are