Sequences
Sequences covers arithmetic and geometric sequences, sums, recurrence relations, and mathematical induction. The exam tests both direct formulas and the ability to transform a recurrence into a familiar sequence.
Core rules
- Arithmetic: a_n=a_1+(n-1)d; S_n=n(a_1+a_n)/2.
- Geometric: a_n=a_1 r^{n-1}; S_n=a_1(1-r^n)/(1-r) for r≠1.
- A recurrence such as a_n=pa_{n-1}+q can sometimes be shifted: a_n-α=p(a_{n-1}-α) when α=q/(1-p).
- Induction: prove the base case, assume the statement for n=k, and use it to prove n=k+1.
- For sums, count the number of terms as last index minus first index plus one.
Worked example
Solve a_1=1, a_n=2a_{n-1}+1. Add 1: a_n+1=2(a_{n-1}+1). Thus a_n+1 is geometric with first term 2 and ratio 2, so a_n+1=2^n. Hence a_n=2^n-1.
Traps
Check whether a sequence starts at n=0 or n=1. In a geometric sum, the formula changes when r=1. Do not use the arithmetic sum formula for a geometric sequence. In induction, the base case and the transition must both be shown; assuming the result for all n is circular.