Sequences, series and discrete mathematics
What this domain tests
Arithmetic and geometric sequences, summation, convergence, proof by induction and the binomial theorem. The exam tests both formula recall and the ability to set up a sequence from a worded or recursive description.
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 not equal to 1; the sum to infinity is u_1/(1 - r) only when |r| < 1.
- Convergence: a sequence converges if it approaches a finite limit. Geometric sequences converge only when |r| < 1 (or r = 1).
- Induction: prove the statement for the base case, assume it for n = k, then prove it for n = k + 1, and state the conclusion.
- Binomial theorem: (a + b)^n = sum from k = 0 to n of C(n, k) a^(n-k) b^k; the general term is C(n, k) a^(n-k) b^k.
- Sigma notation: sum from k = 1 to n of u_k means u_1 + u_2 + ... + u_n; split sums and factor constants.
Worked example
The sum of the first n terms of an arithmetic sequence is S_n = 5n^2 + 3n. Find the first term and the common difference. First, u_1 = S_1 = 5 + 3 = 8. Next, S_2 = 20 + 6 = 26, so u_2 = S_2 - S_1 = 26 - 8 = 18, and d = 18 - 8 = 10. Check with the formula: S_n = (n/2)(2(8) + (n - 1)10) = (n/2)(10n + 6) = 5n^2 + 3n, which matches.
Traps that cost marks
- Using n instead of n - 1 in the nth-term formula, or mixing up u_n and S_n.
- Applying the sum-to-infinity formula when |r| >= 1.
- An induction proof that omits the base case, or that assumes the result for n = k + 1.
- Errors in binomial coefficients, particularly forgetting that C(n, k) = C(n, n - k).