Sequences, series and recurrence · Extension
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| common ratio/ˈkɒmən ˈreɪʃɪəʊ/ | công bội |
Does the change add or multiply?
- A saving plan adds ¥30 more each week; a population model grows by 5% each year. Equal differences and equal ratios need different models.
- This lesson studies common ratio 公比: The constant multiplier between consecutive terms of a geometric sequence.
The method
- For an arithmetic progression, u_n=a+(n-1)d and S_n=n(2a+(n-1)d)/2.
- For a geometric progression, u_n=ar^(n-1) and S_n=a(1-r^n)/(1-r).
- An infinite geometric sum exists only if |r|<1.
An infinite geometric series has a finite sum only when which condition holds?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The terms ar^(n-1) shrink towards zero only when |r|<1, so the sum converges to a/(1-r). With |r|≥1 the terms do not shrink and the sum diverges.
A worked case
For a=5,d=3,n=8, u_8=5+7×3=26 and S_8=8(10+21)/2=124. For a=12,r=1/2, S infinity=12/(1-1/2)=24. For u_(n+1)=2u_n+1 with u_1=1, the next terms are 3,7,15.
Sequences, series and recurrence
For an arithmetic progression, u_n=a+(n-1)d and S_n=n(2a+(n-1)d)/2
Compare the model with the worked case and explain one change.
For a=5,d=3, find u_8.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
u₈=5+(8-1)×3=26.
Where it goes wrong
- The first term has index 1, so the exponent is n-1.
- A sequence is a list; a series is a sum.
- A geometric sequence can alternate in sign and still converge.
Not true: Every geometric series has a finite sum to infinity. A sum to infinity exists only when −1<r<1; for r=2 the terms grow without limit.
Find the sum of the first 8 terms with a=5,d=3.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
S₈=8[2×5+7×3]/2=124.
A sum to infinity exists only when −1<r<1.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
True. A sum to infinity exists only when −1<r<1; for r=2 the terms grow without limit.
Applying it
- Explain whether the context justifies additive or multiplicative change.
- In finance, distinguish a single deposit from a stream of deposits before choosing a sum formula.
Find the infinite sum with a=12,r=0.5.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Since |r|<1, S∞=12/(1-0.5)=24.
Match each formula to the quantity it gives.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Arithmetic sequences add a constant difference; geometric sequences multiply by a constant ratio; only a shrinking ratio gives a finite infinite sum.
In your course
- OxfordAQA International GCSE Mathematics (9260), Extension tier, specification section 3.2.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Common ratio: The constant multiplier between consecutive terms of a geometric sequence.
- For an arithmetic progression, u_n=a+(n-1)d and S_n=n(2a+(n-1)d)/2.
- Watch for: the first term has index 1, so the exponent is n-1.
Write from a five-term sum of 80
- An arithmetic series has first term 4, difference 6 and 5 terms.
- Find the fifth term and the sum of the five terms.
An arithmetic series has first term 4, common difference 6 and 5 terms. What are the fifth term and the sum of these five terms?
Five times 32 divided by 2 is 80. The formula uses the first term and the difference.
What is the sum of the five terms?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
5/2 times 32 is 80. Other options — 124 (the lesson sum of eight terms of a different series); 28 (the fifth term rather than the sum).