Number and algebra
What Number and algebra tests
This domain tests structure and manipulation: arithmetic and geometric sequences and series, growth and decay, logarithms, and the binomial expansion. The exam rewards choosing the right formula, respecting domains, and interpreting the result in context rather than only computing.
Core rules
- Arithmetic: u_n = u_1 + (n − 1)d; S_n = n/2(2u_1 + (n − 1)d) = n/2(u_1 + u_n).
- Geometric: u_n = u_1 r^(n−1); S_n = u_1(1 − r^n)/(1 − r) for r ≠ 1; S∞ = u_1/(1 − r) only when |r| < 1.
- Financial models: compound interest A = P(1 + r)^n; depreciation A = P(1 − r)^n; match the rate and the number of compounding periods.
- Logarithms: log_a x = y iff a^y = x; log_a(xy) = log_a x + log_a y; log_a(x^k) = k log_a x; log_a b = ln b / ln a.
- Binomial: (a + b)^n = sum from r = 0 to n of C(n,r) a^(n−r) b^r, where C(n,r) = n!/(r!(n−r)!).
Worked example
A geometric series has u_1 = 8 and r = 0.75. Find S∞ and the least n with S_n > 30. S∞ = 8/(1 − 0.75) = 32. Also S_n = 32(1 − 0.75^n). Need 32(1 − 0.75^n) > 30, so 0.75^n < 1/16. Taking logs: n > ln(1/16)/ln(0.75) ≈ 9.64, so n = 10. Check: S_9 ≈ 29.60, S_10 ≈ 30.20.
Traps
Mixing n and n − 1; using S∞ when |r| ≥ 1; taking logs of non-positive numbers; forgetting a log domain; rounding before the final answer; in binomial, confusing the term number with the power of b.