Sequences, finite sums, exponentials and logarithms
| English | 中文 | Pinyin |
|---|---|---|
| logarithm/ˈlɒɡərɪθəm/ | 对数 | duì shù |
| common ratio/ˈkɒmən ˈreɪʃɪəʊ/ | 公比 | gōng bǐ |
A decision before an answer
- The fifth term of a sequence uses four steps after the first. Confusing a term number with the number of intervals gives an off-by-one answer.
- Your goal: Distinguish arithmetic difference from geometric ratio.
Read the relationship
- An arithmetic sequence has constant difference d: aₙ=a₁+(n-1)d. Its finite sum is n(a₁+aₙ)/2. A geometric sequence has constant ratio r: aₙ=a₁r^(n-1). For r≠1, a finite sum is a₁(1-rⁿ)/(1-r); when r=1 it is simply n a₁. Match the requested term or total.
- Find a term or finite sum while keeping indexing consistent.
Arithmetic sequence starts at 5 with difference 4. Fourth term:
5+3·4=17.
Use the defining rule
- An exponential model A b^t represents multiplication by b per stated time unit. A percentage growth r uses factor 1+r; repeated changes multiply factors rather than adding percentages. If doubling takes several time units, divide t by that interval in the exponent.
- Convert logarithmic equations to exponential form on the valid domain.
log₃ 81 equals:
3⁴=81.
Check the conditions
- For base b>0 with b≠1, log_b(a)=c means b^c=a and requires a>0. Solve a logarithmic equation by translating it to an exponential relationship, then check the argument. Logarithms of products add under valid positive arguments; logarithms do not distribute over addition.
- Convert logarithmic equations to exponential form on the valid domain.
Arithmetic 4,7,10,… has a₅=16 and sum of first five=5(4+16)/2=50. Geometric 3,6,12,24 has four-term sum 45. log₂(x-1)=4 gives x-1=16, so x=17, which satisfies x>1. A population 50·2^(t/3) reaches 200 after six time units.
The first three terms 2,6,18 have sum ____.
Add the three terms; 18 alone is the last term.
Apply the task format
- Use exact powers when available. ACT may test a simple logarithmic relationship without requiring numerical logarithm tables. A calculated approximation must still respect the question’s representation and domain. Keep a sequence’s index, an exponential model’s time and a logarithm’s argument conceptually separate.
- Convert logarithmic equations to exponential form on the valid domain.
Do not use n rather than n-1 for the nth term, confuse a finite sum with its last term, or expand log(a+b) as log a + log b.
Which answer fits this case?
Distinguish arithmetic difference from geometric ratio
log(a+b)=log a+log b for all positive a,b.
The product rule concerns ab, not a+b.
Keep the distinctions
- common ratio 公比 — The fixed multiplier between consecutive geometric terms.
- logarithm 对数 — The exponent needed to obtain an argument from a stated base.
- Distinguish arithmetic difference from geometric ratio.
- Find a term or finite sum while keeping indexing consistent.
- Convert logarithmic equations to exponential form on the valid domain.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.