1. **Problem Statement:** Calculate simple interest, compound interest, loan repayments, and investment growth using arithmetic progression (AP) and geometric progression (GP). 2. **Simple Interest (AP):** Simple interest is calculated using the formula $$SI = P \times r \times t$$ where $P$ is the principal, $r$ is the rate per period, and $t$ is the time. The total amount after $t$ periods is $$A = P + SI = P(1 + rt)$$. This forms an arithmetic progression because interest added each period is constant. 3. **Compound Interest (GP):** Compound interest is calculated using $$A = P(1 + r)^t$$ where $P$ is the principal, $r$ is the interest rate per period, and $t$ is the number of periods. The amount grows geometrically because interest is added on the accumulated amount each period. 4. **Loan Repayments:** For loans with fixed payments, the repayment amount can be found using the annuity formula $$R = P \frac{r(1+r)^n}{(1+r)^n - 1}$$ where $R$ is the regular payment, $P$ is the loan amount, $r$ is the interest rate per period, and $n$ is the number of payments. This formula derives from the sum of a geometric series. 5. **Investments:** Investment growth with reinvested interest follows compound interest formula $$A = P(1 + r)^t$$. For regular contributions, the future value of an annuity formula applies: $$FV = PMT \frac{(1+r)^n - 1}{r}$$ where $PMT$ is the payment per period. 6. **Summary:** Use AP formulas for simple interest where interest is linear over time. Use GP formulas for compound interest and loan repayments where interest compounds or payments form a geometric series. This explanation covers the key formulas and concepts for financial applications of arithmetic and geometric progressions.