Linear growth and exponential growth are two fundamental patterns of growth that differ significantly in their characteristics and rates of change:

Linear Growth:

1. Constant Increase: In linear growth, a fixed amount is added in each time period. The growth rate remains constant over time.
   • Example: A sequence increasing by 2 units per step (e.g., 2, 4, 6, 8, ...).
2. Mathematical Model: The general formula for linear growth is: $y = mx + c$ where $m$ is the rate of change, $x$ is the time or independent variable, and $c$ is the initial value.
3. Graphical Representation: The graph of linear growth is a straight line.
4. Rate of Growth: The growth is additive; it does not accelerate.
5. Applications: Often seen in scenarios with fixed-rate processes, such as saving a fixed amount of money every month.

---

Exponential Growth:

1. Proportional Increase: In exponential growth, the quantity grows by a constant factor (or percentage) in each time period. The growth rate increases over time because it depends on the current value.
   • Example: A sequence doubling each step (e.g., 2, 4, 8, 16, ...).
2. Mathematical Model: The general formula for exponential growth is: $y = a \cdot b^x$ where $a$ is the initial value, $b$ is the growth factor (e.g., 2 for doubling), and $x$ is the time or independent variable.
3. Graphical Representation: The graph of exponential growth is a curve that starts slow but rises steeply.
4. Rate of Growth: The growth is multiplicative; it accelerates as time progresses.
5. Applications: Common in natural processes, such as population growth, compound interest, and the spread of diseases.

---

Key Differences:

Feature	Linear Growth	Exponential Growth
Rate of Change	Constant	Increasing
Pattern	Additive	Multiplicative
Graph	Straight Line	Exponential Curve
Formula	$y = mx + c$	$y = a \cdot b^x$
Growth Speed	Steady	Rapid (after initial phase)

Would you like further clarification or examples?

---

Relative Questions:

1. How can we calculate the point where exponential growth surpasses linear growth?
2. What are some real-world examples of linear and exponential growth?
3. How does the graph of linear growth compare to the early stages of exponential growth?
4. What happens when the growth factor in exponential growth is less than 1?
5. How do compound interest formulas illustrate exponential growth?

---

Tip:

When analyzing growth patterns, check the rate of change. If it’s constant, it’s linear; if it’s proportional to the current value, it’s exponential.