Question

(1) April received an inheritance from her grandmother in the form of an annuity. The annuity pays $3,000 on January 1st from 1966 through 1984. Find the value of this annuity on January 1, 1966 using an annual effective interest rate of 5% and represent this value by an appropriate annuity symbol. (2) Suppose i = 3%. Find the value one month before the first payment of a level annuity-due paying $200 at the beginning of each month for five years. (3) (a) Describe in words what the difference a is measuring. a IS measurin (b) Given thatay177208656 andän+I-185248436, find the integer n.

Answer 1

(1) To find the value of the annuity on January 1, 1966, use the present value formula to calculate the present value of the annuity. The value is $43,786.49 represented by the annuity symbol $43,786.49.

(2) To find the value one month before the first payment of the level annuity-due, use the present value formula for a level annuity-due. The value is $10,165.14.

(3) The difference 'a' is measuring the numerical difference between two numbers. Given the equation, the value of 'n' is 362457091.

Step-by-step explanation:

(1) To find the value of the annuity on January 1, 1966, we need to calculate the present value of the annuity. The formula for present value of an annuity is:

PV = PMT × ((1 - (1+r)^-n)/r)

Where PV is the present value, PMT is the payment per period, r is the interest rate per period, and n is the number of periods. In this case, the payment per period is $3,000, the interest rate per period is 5%, and the number of periods is 19 (1966-1984). Plugging these values into the formula, we get:

PV = $3,000 × ((1 - (1+0.05)^-19)/0.05)

PV = $43,786.49

Therefore, the value of the annuity on January 1, 1966 is $43,786.49 represented by the annuity symbol $43,786.49.

(2) To find the value one month before the first payment of the level annuity-due, we need to calculate the present value of the annuity. The formula for present value of a level annuity-due is:

PV = PMT × ((1 - (1+r)^-(n+1))/(1+r))

Where PV is the present value, PMT is the payment per period, r is the interest rate per period, and n is the number of periods. In this case, the payment per period is $200, the interest rate per period is 3%, and the number of periods is 60 (12 months × 5 years). Plugging these values into the formula, we get:

PV = $200 × ((1 - (1+0.03)^-(60+1))/(1+0.03))

PV = $10,165.14

Therefore, the value one month before the first payment of the level annuity-due is $10,165.14.

(3) (a) The difference 'a' is measuring the value obtained by subtracting variable 'n' from variable 'y'. It is measuring the numerical difference between two numbers.

(b) Given that 'a' = 177208656 and 'n+I-185248436, we can solve for the integer value of 'n'. Rearranging the equation, we get:

'a' = 'n' + 1 - 185248436

177208656 = 'n' - 185248435

'n' = 362457091

Therefore, the value of 'n' is 362457091.