Derive The formula for an annuity starting from the 1-term return: P+ = Pt-1(1 + r)

To derive the formula for an annuity, we start with the basic idea of compound interest and build towards the annuity formula. The given expression is:

\[
P_t = P_{t-1}(1 + r)
\]

Where:
- \( P_t \) is the principal at time \( t \).
- \( r \) is the interest rate per period.
- \( P_{t-1} \) is the principal at the previous time period.

We will now generalize this for an annuity, which involves regular payments \( A \) made at the end of each period for \( n \) periods.

### Step 1: Expression for the first few terms of the annuity
At time \( t = 0 \), assume the initial principal \( P_0 = 0 \) (we are considering only payments made). After the first payment \( A \) at the end of period 1, the future value of this payment at time \( t = n \) (after \( n \) periods) is:



Derive the formula for an annuity starting from the 1-term return: P_t = P_{t-1}(1 + r)

Explore the step-by-step derivation of the annuity formula starting from the 1-term return equation: P_t = P_{t-1}(1 + r). This detailed guide covers key concepts such as compound interest and the annuity formula, suitable for college-level students.

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