Number and Algebra
What the domain tests
Number and algebra in AI HL rewards interpretation as much as manipulation: standard form and error, arithmetic and geometric sequences, financial mathematics, and exponential/logarithmic models. Paper 1 expects exact reasoning without a calculator; Paper 2 expects efficient GDC use, including the finance solver. Define variables and state units.
Core rules
- Arithmetic: u_n = u_1 + (n-1)d; S_n = n/2(2u_1+(n-1)d).
- Geometric: u_n = u_1 r^(n-1); S_n = u_1(1-r^n)/(1-r), r≠1; S∞ = u_1/(1-r) for |r|<1.
- Compound growth: FV = PV(1+i)^n, where i is the rate per compounding period and n the number of periods.
- Logarithms solve for an exponent: if a^x=b, x=log_a b = ln b / ln a.
- Annuity and amortization problems use the finance solver; identify PV, PMT, FV, n and I%.
Worked example
$5000 is invested at 4.2% p.a. compounded monthly. How long until it reaches $7000? Here i=0.042/12=0.0035, so 7000=5000(1.0035)^n. Then 1.4=(1.0035)^n, giving n=ln1.4/ln1.0035≈96.3 months. Convert: 96.3/12≈8.03 years. Check: 96 months gives about $6993, so 97 months is needed to actually reach $7000.
Traps
Using the annual rate as the monthly rate; entering years instead of compounding periods; rounding the monthly rate before solving; confusing nominal and effective annual rates; forgetting that an amortizing loan has FV=0 and payments are negative in the solver; using S∞ when |r|≥1.