Sum of a Geometric Series - Annuity Formula
The future value of an annuity (regular payments into an account earning compound interest) forms a geometric series. For a payment P made at the end of each compounding period, with interest rate r compounded n times per year, the total value A_t after t years is: A_t = \frac{P \left( \left(1 + \frac{r}{n}\right)^{nt} - 1 \right)}{\frac{r}{n}} This formula represents the sum of a finite geometric series where each payment grows by the factor \left(1 + \frac{r}{n}\right). An annuity creates a geometric series because each payment earns compound interest for a different amount of time. This skill is part of Grade 11 math in enVision, Algebra 2.
Key Concepts
The future value of an annuity (regular payments into an account earning compound interest) forms a geometric series. For a payment $P$ made at the end of each compounding period, with interest rate $r$ compounded $n$ times per year, the total value $A t$ after $t$ years is:.
$$A t = \frac{P \left( \left(1 + \frac{r}{n}\right)^{nt} 1 \right)}{\frac{r}{n}}$$.
Common Questions
What is Sum of a Geometric Series - Annuity Formula?
The future value of an annuity (regular payments into an account earning compound interest) forms a geometric series. For a payment P made at the end of each compounding period, with interest rate r compounded n times per year, the total value A_t after t years is: A_t = \frac{P \left( \left(1 + \frac{r}{n}\right)^{nt} - 1 \right)}{\frac{r}{n}} This formula represents the sum of a finite geometric series where each payment grows by the factor \le
How does Sum of a Geometric Series - Annuity Formula work?
Example: Monthly payments of 200 at 6% annual interest compounded monthly for 10 years: $A_{10} = \frac{200 \left( \left(1 + \frac{0.06}{12}\right)^{12 \cdot 10} - 1 \right)}{\frac{0.06}{12}} \approx 32,776.77$ dollars.
Give an example of Sum of a Geometric Series - Annuity Formula.
Quarterly payments of 1000 at 8% annual interest compounded quarterly for 15 years: $A_{15} = \frac{1000 \left( \left(1 + \frac{0.08}{4}\right)^{4 \cdot 15} - 1 \right)}{\frac{0.08}{4}} \approx 108,648.23$ dollars.
Why is Sum of a Geometric Series - Annuity Formula important in math?
An annuity creates a geometric series because each payment earns compound interest for a different amount of time. The first payment earns interest for the longest period, the second payment for one period less, and so on.
What grade level covers Sum of a Geometric Series - Annuity Formula?
Sum of a Geometric Series - Annuity Formula is a Grade 11 math topic covered in enVision, Algebra 2 in Chapter 6: Exponential and Logarithmic Functions. Students at this level study the concept as part of their grade-level standards and are expected to explain, analyze, and apply what they have learned.
What are typical Sum of a Geometric Series - Annuity Formula problems?
Monthly payments of 200 at 6% annual interest compounded monthly for 10 years: $A_{10} = \frac{200 \left( \left(1 + \frac{0.06}{12}\right)^{12 \cdot 10} - 1 \right)}{\frac{0.06}{12}} \approx 32,776.77$ dollars.; Quarterly payments of 1000 at 8% annual interest compounded quarterly for 15 years: $A_{15} = \frac{1000 \left( \left(1 + \frac{0.08}{4}\right)^{4 \cdot 15} - 1 \right)}{\frac{0.08}{4}} \approx 108,648.23$ dollars.; Annual payments of 500