Math Problem Statement

Question content area top Part 1 ​(Compound annuity​) What is the accumulated sum of each of the following streams of​ payments? a. ​$510 a year for 9 years compounded annually at 9 percent. b. ​$104 a year for 6 years compounded annually at 8 percent. c. ​$38 a year for 12 years compounded annually at 11percent. d. ​$23 a year for 5 years compounded annually at 4 percent. Question content area bottom Part 1 a. What is the accumulated sum of ​$510 a year for 9 years compounded annually at 9 ​percent? ​$    enter your response here  ​(Round to the nearest​ cent.)

Solution

To calculate the accumulated sum of a stream of payments (a compound annuity), we use the future value of an annuity formula:

FV=P×(1+r)n1rFV = P \times \frac{(1 + r)^n - 1}{r}

Where:

  • FVFV = Future Value
  • PP = Annual payment
  • rr = Interest rate per period (as a decimal)
  • nn = Number of years

Given for part (a):

  • P=510P = 510
  • r=9%=0.09r = 9\% = 0.09
  • n=9n = 9

Now, we compute:

FV=510×(1.09)910.09FV = 510 \times \frac{(1.09)^9 - 1}{0.09}

Let's calculate. Given values for part (a) P = 510 # Annual payment r = 0.09 # Interest rate per year n = 9 # Number of years

Future Value calculation

FV_a = P * ((1 + r)**n - 1) / r FV_aThe accumulated sum for part (a) is $6,640.73 (rounded to the nearest cent).

Would you like me to proceed with the other parts (b, c, d)?

Here are five related questions to deepen understanding:

  1. How does increasing the interest rate affect the accumulated sum?
  2. What happens if payments are made semi-annually instead of annually?
  3. How does the number of years impact the future value of an annuity?
  4. Can this formula be applied to monthly deposits? If so, how?
  5. What is the difference between a future value annuity and a present value annuity?

Tip: The longer the time period and the higher the interest rate, the more significant the compound growth effect!

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Math Problem Analysis

Mathematical Concepts

Compound Annuity
Future Value of Annuity
Interest Calculations

Formulas

FV = P * ((1 + r)^n - 1) / r

Theorems

Future Value of Annuity Theorem

Suitable Grade Level

Grades 11-12