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. These exercises reward 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 (a>0, a≠1, positive arguments): 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.
Proof and complex numbers
An induction proof establishes a base case and then derives the next case from an assumed arbitrary case; checking several examples alone is not induction. For a complex number z=a+bi, its conjugate is a-bi and z times its conjugate is a²+b². A polynomial with real coefficients has nonreal roots in conjugate pairs. In polar form, multiplication multiplies moduli and adds arguments. State the chosen argument interval when reporting roots or angles, and distinguish a complex modulus from its real part.