$1.$

$(1)$ Obtain the linear trend equation for the following data on new checking accounts at Fair Saving Bank and use it to predict new checking accounts for periods $16$ to $19.$

$\backslash$begin$\{$tabular$\}\{|$r$|$r$|$r$|$r$|$r$|$r$|\}$

$\backslash$hline $\backslash$multicolumn$\{1\}\{|$c$|\}\{$ Period $\}$ & New Accounts & $\backslash$multicolumn$\{1\}\{$c$|\}\{$ Period $\}$ & New Accounts & Period & New Accounts $\backslash \backslash$

$\backslash$hline $1$ & $200$ & $6$ & $232$ & $11$ & $281\backslash \backslash$

$\backslash$hline $2$ & $214$ & $7$ & $248$ & $12$ & $275\backslash \backslash$

$\backslash$hline $3$ & $211$ & $8$ & $250$ & $13$ & $280\backslash \backslash$

$\backslash$hline $4$ & $228$ & $9$ & $253$ & $14$ & $288\backslash \backslash$

$\backslash$hline $5$ & $235$ & $10$ & $267$ & $15$ & $310\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$

$(2)$ Use trend$-$adjusted exponential smoothing with $\backslash ($ S$\_\{4\}=228,$ G$\_\{4\}=9\mathrm{.}33,\backslash$alpha$=0\mathrm{.}3\backslash )$ and $\backslash (\backslash$beta$=0\mathrm{.}2\backslash )$ to smooth the new account data in part $(1).$ What is the forecast for period $16?$

provide the solutions with answer, thank you!