AQA A-level Mathematics 7357, section D (Sequences and series), version 1.3 (first assessment 2018). Original teaching notes; objective-level mapping lives in content/teaching/aqa_a_level_mathematics/objective-review.yaml.
Prerequisites: GCSE arithmetic sequences and simultaneous equations. Each lesson develops its own method before independent transfer.
These are original teaching examples, not official AQA material.
A saver deposits the same amount every year while an investment grows by the same percentage every year. After eight years, which balance can you total exactly in one line?
Start with a prediction. State which of the two grows by addition and which by multiplication, and what each sum formula needs.
Build the model
An arithmetic sequence 等差数列 adds a constant difference: uₙ = a + (n − 1)d. Its series sums by pairing ends: Sₙ = n(a + uₙ)/2. A geometric sequence 等比数列 multiplies by a constant ratio: uₙ = arⁿ⁻¹, with Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1.
An infinite geometric sum exists only when |r| < 1, because the terms shrink towards zero: S∞ = a/(1 − r). A recurrence relation 递推关系 defines each term from the previous one and needs an initial value; a periodic sequence 周期数列 repeats after a fixed number of steps.
arithmetic sequence: A sequence with a constant difference between terms; geometric sequence: A sequence with a constant ratio between terms; recurrence relation: A rule giving each term from the preceding value; periodic sequence: A sequence whose terms repeat after a fixed number of steps.
Choose evidence that can test it
Recover a and d from two stated terms by simultaneous equations before using a formula: u₃ = 10 and u₇ = 22 give 4d = 12, so d = 3 and a = 4. A term count must come out as a positive integer; a fraction or negative root is rejected, never rounded into place.
Check a claimed sum against the first few terms directly, and verify an at-least target against the integers on BOTH sides of the crossing.
Work from known quantities
State the known values and their units. Choose the relation because its assumptions fit this case, then rearrange before substitution.
Known: a = 5, d = 3, n = 8. The eighth term is u₈ = 5 + 7 × 3 = 26. The sum is S₈ = n(a + uₙ)/2 = 8(5 + 26)/2 = 124. Each of the four end-pairs totals 31, and 4 × 31 = 124 ✓.
Example:
For a = 12, r = 0.5, find the infinite sum and the first n whose partial sum exceeds 23.9, using the same sequence: known → formula → substitution → integer check. (Answer: S∞ = 12/(1 − 0.5) = 24. The partial sums are Sₙ = 24(1 − 0.5ⁿ): S₇ = 24(1 − 1/128) = 23.8125 < 23.9 and S₈ = 24(1 − 1/256) = 23.90625 ≥ 23.9, so n = 8.)
Check the conclusion and its limits
The infinite-sum formula is invalid at r = 1 or r = −1, where the terms do not shrink; a negative ratio converges if its MAGNITUDE is below 1. A recurrence needs the operation order preserved exactly: B next = 1.1B + 50 differs from 1.1(B + 50).
Return to the original observation. Explain what the result supports, which conditions it assumes, and one way to test a competing explanation.
Warn:
Every geometric series has a finite sum to infinity. This claim is false: the sum exists only when |r| < 1. At r = 2 the terms grow without bound, and at r = −1 the partial sums oscillate forever — neither converges.
Key:
Sequences and series: arithmetic adds (pair the ends to sum), geometric multiplies (Sₙ = a(1−rⁿ)/(1−r), and S∞ = a/(1−r) only for |r| < 1). Recover parameters from stated terms, and demand integer answers for counts.
