Mimi is planning to save for the next 48 months in order to raise a deposit to buy a flat She plans to save \( \$ 300 \) this month and each successive month thereafter, to save an extra \( \$ 5 \) compared to the previous month. a) Find the amount she will save on the twelfth month. b) Find the total amount she will save at the end of the 48 months. Emmanuel is also planning to save for the next 48 months in order to buy a car. He plans to save \( \$ \) a this month and each successive month thereafter, to save an extra \( \$ 15 \) compared to the previous month. c) Find the value of \( a \), if Emmanuel saves the same amount of money as Anita does in the next 48 months.

To find out how much Mimi saves in the twelfth month, we can use the formula for an arithmetic sequence. The first month she saves $300, and she increases her savings by $5 each month. Thus, in the twelfth month, she saves: \( 300 + (12 - 1) \times 5 = 300 + 55 = \$355. \) For the total amount saved over 48 months, we can calculate the total of the arithmetic sequence where \( n = 48\), the first term \( a = 300 \), and the last term \( l \) is: \( l = 300 + (48 - 1) \times 5 = 300 + 235 = 535. \) The sum \( S \) of the first \( n \) terms is given by: \( S_n = \frac{n}{2} \times (a + l) \) So, \( S_{48} = \frac{48}{2} \times (300 + 535) = 24 \times 835 = \$20,040. \) Now, for Emmanuel, since he wants to save the same total amount in 48 months as Mimi, we need to set his annual increase and the resulting total equal to Mimi's. Let Emmanuel's first month's saving be \( a \). The amount he saves in the 48th month will be: \( a + (48 - 1) \times 15 = a + 705. \) His total savings can be calculated in a similar manner: \( S_{48} = \frac{48}{2} \times (a + (a + 705)) = 24 \times (2a + 705) = 48a + 16,920. \) Equating Emmanuel's savings to Mimi's, we get: \( 48a + 16,920 = 20,040. \) Solving for \( a \): \( 48a = 20,040 - 16,920 = 3,120 \) \( a = \frac{3,120}{48} = 65. \) So Emmanuel's first month's saving should be \( a = \$65. \)