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. :

We’re given a few arithmetic sequences. Let’s go through each part step by step. ───────────────────────────── Part (a): Mimi’s Savings for the 12th Month Mimi saves $300 in the first month and increases her savings by $5 each month. This forms an arithmetic sequence with:  First term, a₁ = 300  Common difference, d = 5 The formula for the nth term of an arithmetic sequence is:  aₙ = a₁ + (n – 1) * d For the 12th month (n = 12):  a₁₂ = 300 + (12 – 1) * 5     = 300 + 11 * 5     = 300 + 55     = 355 So, Mimi saves $355 in the 12th month. ───────────────────────────── Part (b): Total Savings Over 48 Months The sum of the first n terms of an arithmetic sequence is given by:  Sₙ = n/2 * [2a₁ + (n – 1) * d] For 48 months (n = 48):  S₄₈ = 48/2 * [2(300) + (48 – 1) * 5]    = 24 * [600 + 47 * 5]    = 24 * [600 + 235]    = 24 * 835    = 20,040 So, Mimi’s total savings at the end of 48 months is $20,040. ───────────────────────────── Part (c): Emmanuel’s Required First Month Savings Emmanuel plans to save for 48 months too. His savings also form an arithmetic sequence:  Let his first month’s saving be a (unknown)  He increases his savings by $15 each month, so d = 15 His total savings for 48 months is given by:  S₄₈ = 48/2 * [2a + (48 – 1) * 15]    = 24 * [2a + 47 * 15] We know from Part (b) that he must save the same total as Mimi, which is $20,040. Therefore, set up the equation:  24 * [2a + (47 × 15)] = 20,040 First, calculate 47 × 15:  47 × 15 = 705 Now the equation is:  24 * (2a + 705) = 20,040 Divide both sides by 24:  2a + 705 = 20,040 / 24  2a + 705 = 835 Subtract 705 from both sides:  2a = 835 – 705  2a = 130 Divide by 2:  a = 130 / 2  a = 65 Thus, Emmanuel must start with $65 in the first month. ───────────────────────────── Final Answers a) Mimi saves $355 on the 12th month. b) Mimi saves a total of $20,040 over 48 months. c) Emmanuel’s first month saving amount, a, is $65.