Overview
This lecture introduces modeling compound interest using recursive relations, highlighting differences between simple and compound interest, and providing step-by-step examples for various compounding scenarios.
Introduction to Compound Interest & Sequences
- Simple interest adds a constant amount each year (arithmetic sequence).
- Compound interest reinvests earned interest each year, so each year's interest increases (geometric sequence).
- Geometric sequences multiply each term by a constant ratio, unlike arithmetic sequences with a constant difference.
Recursive Relations for Interest Calculations
- Recursive relations repeatedly use previous values to calculate the next balance.
- Simple interest recursion: ( V_{n+1} = V_n + d ), where ( d ) is the fixed interest amount.
- Compound interest recursion: ( V_{n+1} = r \times V_n ), where ( r ) is the growth factor (1 + interest rate as a decimal).
Example Calculations: Compound Interest
- For $1,000 at 8% interest:
- Initial: ( V_0 = 1,000 ), ( r = 1.08 )
- Year 1: ( V_1 = 1.08 \times 1,000 = 1,080 )
- Year 2: ( V_2 = 1.08 \times 1,080 = 1,166.40 )
- Year 3: ( V_3 = 1.08 \times 1,166.40 = 1,259.71 )
- To find when the investment exceeds $1,500, repeat calculations until ( V_n > 1,500 ) (after 6 years).
Compound Frequency Adjustments
- When compounding frequency changes (yearly, quarterly, monthly), adjust the rate:
- Quarterly: divide annual rate by 4; monthly: divide by 12.
- Example: $5,000 loan at 4.5% annually, compounded quarterly:
- Quarterly rate = 4.5% ÷ 4 = 1.125%
- Recursive relation: ( V_{n+1} = 1.01125 \times V_n ) with ( V_0 = 5,000 )
Practice Problem Approach
- State principal and define what ( V_n ) represents (e.g., after n quarters).
- Adjust interest rate for compounding period.
- For "after 1 year" with quarterly compounding, calculate ( V_4 ).
Key Terms & Definitions
- Recursive Relation — An equation using previous terms to find the next in a sequence.
- Principal — The initial amount of money invested or borrowed.
- Geometric Sequence — A sequence where each term is multiplied by the same ratio.
- Compound Frequency — How often interest is applied (yearly, quarterly, etc.).
- Growth Factor (r) — ( 1 + \text{interest rate (as decimal)} ) per period.
Action Items / Next Steps
- Complete the last two practice questions assigned in class, applying recursive relations for quarterly compounding.