- 1. What Makes a Sequence?
Name the terms of a sequence with U_n notation, tell finite from infinite sequences, find and use a term-to-term rule, use a formula for the nth term forwards and backwards, and sort sequences by testing for a common difference or a common ratio.
- 2. The Nth Term of a Linear Sequence (A.P.)
Derive and use U_n = a + (n - 1)d for an arithmetic progression: find any term, count the terms, find which term a value is, find the A.P. from two of its terms, and use 2y = x + z for three consecutive terms.
- 3. The Sum of an Arithmetic Series
Derive S_n = n/2 (a + l) by pairing, use S_n = n/2 [2a + (n - 1)d], work backwards from a sum to n, d or a, and solve three-term and real-world problems with arithmetic series.
- 4. The Nth Term of an Exponential Sequence (G.P.)
Derive and use U_n = ar^(n-1) for a geometric progression, work with negative, fractional and surd ratios, find a G.P. from two of its terms (including the plus-or-minus case), and use y^2 = xz for three consecutive terms.
- 5. The Sum of a Finite Geometric Series
Derive S_n = a(r^n - 1)/(r - 1) by shifting and subtracting, use both forms for ratios above and below 1 and for negative ratios, find sums from two given terms, find how many terms give a sum, and choose the correct ratio in increasing or decreasing G.P. problems.
- 6. Sum to Infinity of a G.P.
See why a G.P. with -1 < r < 1 has a finite sum to infinity a/(1 - r), test whether a series converges, work backwards from a sum to infinity to r or a while rejecting roots with |r| >= 1, and apply the idea to recurring decimals and bouncing balls.
- 7. Working Backwards: A Sequence Given by Its Sum
Recover terms and the general term from a formula for S_n using U_n = S_n - S_(n-1) and U_1 = S_1, read a and d straight from S_n = pn^2 + qn, handle a constant term, and recognise a G.P. hidden in a sum formula.
- 8. Real-World Growth and Mixed A.P./G.P. Problems
Model real situations with arithmetic and geometric progressions: savings, costs and training schedules; compound interest, population growth and depreciation; totals over time and sums of a slice of terms; and problems where A.P. terms form a G.P.
- 9. Capstone: Putting It All Together
Choose the right tool for any sequence or series question: recognise an A.P., a G.P. or a sequence given by its sum, translate WAEC's wording, work through multi-part questions, and (as a stretch) generate terms from a recurrence relation.
