Sequences and Series
What the exam tests
Sequences and Series tests arithmetic and geometric patterns, recurrence relations, summation, and mathematical induction. Kyoto problems often define a sequence recursively and ask for a closed form or a limit.
Core rules
- Arithmetic: a_n = a_1 + (n - 1)d, and S_n = n(a_1 + a_n)/2 = n[2a_1 + (n - 1)d]/2.
- Geometric: a_n = a_1 r^{n-1}, and S_n = a_1(1 - r^n)/(1 - r) for r ≠ 1. An infinite geometric series converges exactly when |r| < 1, to a_1/(1 - r).
- For a first-order recurrence a_{n+1} = p a_n + q, try a constant particular solution L = pL + q, then set b_n = a_n - L to get a geometric recurrence.
- Induction: prove the base case, assume the statement for n = k, and use that assumption to prove it for n = k + 1.
Worked example
Let a_1 = 2 and a_{n+1} = 3a_n + 1. Find a closed form. Solve L = 3L + 1, giving L = -1/2. Set b_n = a_n + 1/2. Then b_{n+1} = a_{n+1} + 1/2 = 3a_n + 3/2 = 3(a_n + 1/2) = 3b_n. Also b_1 = 2 + 1/2 = 5/2. Hence b_n = (5/2)3^{n-1}, so a_n = (5/2)3^{n-1} - 1/2.
Traps
Check the starting index before applying a sum formula. A geometric series with r = 1 must be summed as n a_1, not by the fraction formula. Do not assume convergence without |r| < 1. In induction, use the statement for k, not the statement for k + 1.