1. Let's start by stating the problem: We want to understand the difference between an arithmetic sequence and a geometric sequence. 2. An arithmetic sequence is a list of numbers where the same number, called the common difference $d$, is added each time to get the next term. The formula for the $n$th term $a_n$ of an arithmetic sequence is: $$a_n = a_1 + (n-1)d$$ where $a_1$ is the first term. 3. A geometric sequence is a list of numbers where the same number, called the common ratio $r$, is multiplied each time to get the next term. The formula for the $n$th term $a_n$ of a geometric sequence is: $$a_n = a_1 \times r^{n-1}$$ where $a_1$ is the first term. 4. Important rules: - In arithmetic sequences, the difference between consecutive terms is constant. - In geometric sequences, the ratio between consecutive terms is constant. 5. Example of arithmetic sequence: If $a_1 = 3$ and $d = 2$, then the sequence is $3, 5, 7, 9, ...$ and the 5th term is: $$a_5 = 3 + (5-1) \times 2 = 3 + 8 = 11$$ 6. Example of geometric sequence: If $a_1 = 2$ and $r = 3$, then the sequence is $2, 6, 18, 54, ...$ and the 4th term is: $$a_4 = 2 \times 3^{4-1} = 2 \times 27 = 54$$ This explains the difference and formulas for arithmetic and geometric sequences clearly.