Question 1173170
Let's tackle each of these financial calculations step-by-step.

**Question 1: Lump Sum Deposit for Future Value**

1.  **Identify the variables:**
    * Future value (FV): $50,000
    * Interest rate per year (r): 18% or 0.18
    * Compounding frequency (n): Monthly, so 12 times per year
    * Interest rate per period (i): r/n = 0.18 / 12 = 0.015
    * Number of years (t): 6
    * Total number of periods (N): t * n = 6 * 12 = 72

2.  **Use the present value formula:**

    * PV = FV / (1 + i)^N

3.  **Plug in the values:**

    * PV = $50,000 / (1 + 0.015)^72
    * PV = $50,000 / (1.015)^72
    * PV = $50,000 / 2.930491879
    * PV = $17,061.85 (approximately)

Therefore, approximately $17,061.85 must be deposited today.

4.  **Calculate the interest earned:**

    * Interest = FV - PV
    * Interest = $50,000 - $17,061.85
    * Interest = $32,938.15 (approximately)

Therefore, approximately $32,938.15 in interest will be earned.

**Question 3.1: Present Value of an Annuity (Monthly Payments, 12% Interest)**

1.  **Identify the variables:**
    * Payment (PMT): $2,500
    * Number of payments (N): 60
    * Interest rate per year (r): 12% or 0.12
    * Compounding frequency (n): Monthly, so 12 times per year
    * Interest rate per period (i): r/n = 0.12 / 12 = 0.01

2.  **Use the present value of an ordinary annuity formula:**

    * PV = PMT * [(1 - (1 + i)^-N) / i]

3.  **Plug in the values:**

    * PV = $2,500 * [(1 - (1 + 0.01)^-60) / 0.01]
    * PV = $2,500 * [(1 - (1.01)^-60) / 0.01]
    * PV = $2,500 * [(1 - 0.5504495) / 0.01]
    * PV = $2,500 * [0.4495505 / 0.01]
    * PV = $2,500 * 44.95505
    * PV = $112,387.63 (approximately)

Therefore, the present value is approximately $112,387.63.

**Question 3.2: Present Value of an Annuity (Monthly Payments, 18% Interest)**

1.  **Identify the variables:**
    * Payment (PMT): $5,000
    * Number of payments (N): 36
    * Interest rate per year (r): 18% or 0.18
    * Compounding frequency (n): Monthly, so 12 times per year
    * Interest rate per period (i): r/n = 0.18 / 12 = 0.015

2.  **Use the present value of an ordinary annuity formula:**

    * PV = PMT * [(1 - (1 + i)^-N) / i]

3.  **Plug in the values:**

    * PV = $5,000 * [(1 - (1 + 0.015)^-36) / 0.015]
    * PV = $5,000 * [(1 - (1.015)^-36) / 0.015]
    * PV = $5,000 * [(1 - 0.580083) / 0.015]
    * PV = $5,000 * [0.419917 / 0.015]
    * PV = $5,000 * 27.99446
    * PV = $139,972.30 (approximately)

Therefore, the present value is approximately $139,972.30.

**Question 4: Life Insurance Policy (Annuity)**

1.  **Identify the variables:**
    * Payment (PMT): $50,000
    * Number of payments (N): 20
    * Interest rate per year (r): 8% or 0.08
    * Compounding frequency (n): Annually, so 1 time per year
    * Interest rate per period (i): r/n = 0.08 / 1 = 0.08

2.  **Use the present value of an ordinary annuity formula:**

    * PV = PMT * [(1 - (1 + i)^-N) / i]

3.  **Plug in the values:**

    * PV = $50,000 * [(1 - (1 + 0.08)^-20) / 0.08]
    * PV = $50,000 * [(1 - (1.08)^-20) / 0.08]
    * PV = $50,000 * [(1 - 0.2145482) / 0.08]
    * PV = $50,000 * [0.7854518 / 0.08]
    * PV = $50,000 * 9.8181475
    * PV = $490,907.38 (approximately)

(a) Therefore, the amount of insurance should be approximately $490,907.38.

(b) **Calculate the total payments:**

    * Total payments = PMT * N = $50,000 * 20 = $1,000,000

(c) **Calculate the total interest earned:**

    * Total interest = Total payments - PV
    * Total interest = $1,000,000 - $490,907.38
    * Total interest = $509,092.62 (approximately)

Therefore, the interest earned will be approximately $509,092.62.