Sequences and Series: Geometric Progression
Explore **Geometric Progression (G.P.)**, where terms multiply by a **common ratio** ($r$). Observe exponential growth and the concept of an infinite sum.
Help & Instructions
▼- **$n^{th}$ Term:** $a_n = a \cdot r^{n-1}$
- **Infinite Sum ($|r| < 1$):** $S_\infty = \frac{a}{1-r}$
- **Set Parameters:** Enter the starting term ($a$) and the common ratio ($r$).
- **Generate:** Click "Calculate Next Term" to see the series expand.
- **Explore $S_\infty$:** Try a ratio $|r| < 1$ (e.g., $r=0.5$) and see the calculated finite sum for the infinite series!
Define Your Geometric Progression
— Series will appear here —
Series Summary
Current Term ($a_n$): 10
Sum of First $n$ Terms ($S_n$): 10
Infinite Sum ($S_\infty$): —
A **Geometric Progression (G.P.)** is a sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the **common ratio** ($r$). This produces rapid, exponential growth or decay.
The Mathematics Behind the Growth
The sum of an **infinite** G.P. is only possible if the magnitude of the common ratio is less than 1 ($|r| < 1$). If this condition is met, the formula is simple:
$$S_\infty = \frac{a}{1-r}$$If $|r| \ge 1$, the terms continue to grow (or oscillate wildly), and the sum tends toward infinity.
Geometric Progressions are essential in:
- **Finance:** Compound interest and present value calculations.
- **Biology:** Modeling bacterial or viral population growth.
- **Fractals:** Describing geometric patterns that repeat infinitely.
