1. Let's start by defining Arithmetic Sequence (AS) and Geometric Sequence (GS). An Arithmetic Sequence is a sequence of numbers where the difference between consecutive terms is constant, called the common difference $d$. A Geometric Sequence is a sequence where each term is found by multiplying the previous term by a constant called the common ratio $r$. 2. The $n$-th term of an Arithmetic Sequence is given by the formula: $$a_n = a_1 + (n-1)d$$ where $a_1$ is the first term and $d$ is the common difference. 3. The sum of the first $n$ terms of an Arithmetic Sequence (Sum of AS) is: $$S_n = \frac{n}{2} (2a_1 + (n-1)d)$$ This formula comes from pairing terms from the start and end of the sequence. 4. The $n$-th term of a Geometric Sequence is: $$a_n = a_1 r^{n-1}$$ where $a_1$ is the first term and $r$ is the common ratio. 5. The sum of the first $n$ terms of a Geometric Sequence (Sum of GS) is: $$S_n = a_1 \frac{1-r^n}{1-r}$$ provided $r \neq 1$. 6. Important rules: - For AS, the difference $d$ is constant. - For GS, the ratio $r$ is constant. - Factorization can be used to simplify expressions in sums. - Percentage changes can be related to GS when $r = 1 + \frac{p}{100}$ where $p$ is the percentage change. 7. Change of subject means rearranging formulas to solve for a different variable, e.g., solving for $d$ or $r$ from the formulas above. 8. Factorization example: To factor $2a_1 + (n-1)d$ in the sum of AS, you can factor out common terms if needed. 9. Percentage example: If a quantity increases by $p\%$ each time, the common ratio in GS is $r = 1 + \frac{p}{100}$. This covers the basics of AS and GS, their summations, and related concepts like index, change of subject, factorization, and percentage.