Question $2$

$($a$)$ A small retail store observes a steady increase in the number of customers over the past few months. The store manager recorded the number of customers served each month, starting from January to June, as follows:

$\backslash$table$[[$Month$,$January,February,March,April,May,June$],[$Customers$,120,140,160,180,200,220]]$

$($i$)$ Using the formula for the $n^{th}$ term, find the number of customers the store served in December if the trend continues.

$(2$ Marks$)$

$($ii$)$ Find the total number of customers the store served from January to December.

$(2$ Marks$)$

$($b$)$ An investor has invested in three types of bonds: Corporate bonds, Government bonds, and Municipal bonds. Let $x,y,$ and $z$ represent the amounts invested in thousands of dollars in each type of bond, respectively. The investor has the following constraints:

The total investment amount is $100$ thousand Ghana Cedis.

The investor allocates three times as much money in Corporate bonds as in Municipal bonds.

The investment in Government bonds should be $30$ thousand Ghana Cedis more than in Corporate bonds.

$($i$)$ Generate the system of linear equations representing the above constraints.

$(3$ Marks$)$

$($ii$)$ Represent the system of equations in a matrix form $Mx=N.$

$(1$ Mark$)$

$(2$ Marks$)$

$($iii$)$ Find the determinant of the matrix $M.$

$(10$ Marks$)$

$($ii$)$ Hence, find the amounts invested in each type of bond.

TOTAL $(20$ MARKS$)$