수열, 유한 합, 지수함수 및 로그함수
도입됨| English | 의미 |
|---|---|
| logarithm/ˈlɒɡərɪθəm/ | 로그 |
| common ratio/ˈkɒmən ˈreɪʃɪəʊ/ | 공비 |
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:
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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:
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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 ____.
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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.
다음 중 이 사례에 부합하는 답안은 무엇입니까?
산술적 차이와 기하학적 비율 구분하기
log(a+b)=log a+log b for all positive a,b.
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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.
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Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
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The order follows the stated process; check each stage before the next.