1. **Stating the problem:** We will explore key topics in IBDP 1 Math SL including standard form, arithmetic and geometric sequences, financial applications of GP and AP, exponents and logarithms, approximation and errors, and annuities. 2. **Standard Form:** This is writing numbers as $a \times 10^n$ where $1 \leq |a| < 10$ and $n$ is an integer. Example: Write 4500 in standard form. $$4500 = 4.5 \times 10^3$$ 3. **Arithmetic Sequence (AP):** A sequence where each term increases by a constant difference $d$. Formula for $n$th term: $$a_n = a_1 + (n-1)d$$ Example: If $a_1=3$ and $d=2$, find $a_5$. $$a_5 = 3 + (5-1) \times 2 = 3 + 8 = 11$$ 4. **Geometric Sequence (GP):** A sequence where each term is multiplied by a constant ratio $r$. Formula for $n$th term: $$a_n = a_1 \times r^{n-1}$$ Example: If $a_1=2$ and $r=3$, find $a_4$. $$a_4 = 2 \times 3^{3} = 2 \times 27 = 54$$ 5. **Financial Applications of GP and AP:** - AP is used for simple interest where interest is constant. - GP is used for compound interest where interest compounds. Example (Compound Interest): Principal $P=1000$, rate $r=5\%$ per year, find amount after 3 years. $$A = P(1 + r)^n = 1000 \times (1 + 0.05)^3 = 1000 \times 1.157625 = 1157.63$$ 6. **Exponents and Logarithms:** - Exponent rules: $a^m \times a^n = a^{m+n}$, $(a^m)^n = a^{mn}$ - Logarithm is the inverse of exponent: if $a^x = b$, then $\log_a b = x$ Example: Solve $2^x = 8$. Since $8 = 2^3$, $x=3$. 7. **Approximation and Errors:** - Absolute error = $|\text{measured value} - \text{true value}|$ - Relative error = $\frac{\text{absolute error}}{\text{true value}}$ Example: True length = 10 cm, measured = 9.8 cm. Absolute error = $|9.8 - 10| = 0.2$ cm Relative error = $\frac{0.2}{10} = 0.02$ or 2% 8. **Annuities:** Regular payments made over time. Formula for future value of annuity (payments $R$, interest rate $i$, periods $n$): $$FV = R \times \frac{(1+i)^n - 1}{i}$$ Example: $R=100$, $i=0.05$, $n=3$. $$FV = 100 \times \frac{(1.05)^3 - 1}{0.05} = 100 \times \frac{1.157625 - 1}{0.05} = 100 \times 3.1525 = 315.25$$ This covers the main concepts with examples for IBDP 1 Math SL.