1. Let's start by stating the problem: We want to understand the difference between an Arithmetic Progression (AP) and a Geometric Progression (GP). 2. An Arithmetic Progression (AP) is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference $d$. 3. The general formula for the $n$th term of an AP is: $$a_n = a_1 + (n-1)d$$ where $a_1$ is the first term. 4. A Geometric Progression (GP) is a sequence where each term after the first is found by multiplying the previous term by a constant called the common ratio $r$. 5. The general formula for the $n$th term of a GP is: $$a_n = a_1 \times r^{n-1}$$ where $a_1$ is the first term. 6. Important rules: - In AP, the difference between terms is constant: $a_{n} - a_{n-1} = d$ - In GP, the ratio between terms is constant: $\frac{a_n}{a_{n-1}} = r$ 7. Example for AP: If $a_1 = 3$ and $d = 2$, then the sequence is $3, 5, 7, 9, ...$ 8. Example for GP: If $a_1 = 2$ and $r = 3$, then the sequence is $2, 6, 18, 54, ...$ 9. To summarize, AP adds a fixed number each time, GP multiplies by a fixed number each time. This explanation covers the basics of AP and GP.