Questions and Answers
What is the formula for a geometric sequence?
an=a1(r)^n-1
What is the formula for exponential growth?
A=p(1+r)^t
What does A=p(1+-r)^t represent?
Exponential Growth/Decay
What is the formula for a geometric series?
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What are you finding when you use the geometric sequence equation?
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What are you finding when you use the geometric series equation?
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Can the geometric series equation be used with sequences where r>1?
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If a Petri dish has bacteria that double every half hour, what is the growth factor every half hour?
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How will you know if a table displays exponential behavior?
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What is the compound interest formula?
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What is the result of a^x * b^x?
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What is the result of (b^n)^m?
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What is the result of a^x * a^y?
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What is the result of (ab)^x?
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What is the formula for the sum of an infinite geometric series?
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When is the only time you can use infinite geometric series?
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Study Notes
Geometric Sequences and Series
- A geometric sequence is defined by the formula: ( a_n = a_1(r)^{n-1} ), where ( a_n ) is the nth term, ( a_1 ) is the first term, ( r ) is the common ratio, and ( n ) is the term number.
- A geometric series sums the terms of a geometric sequence and is calculated with ( S_n = \frac{a_1(1 - r^n)}{1 - r} ), where ( n ) is the number of terms.
- The infinite geometric series applies when ( r < 1 ) and is calculated with ( S = \frac{A_1}{1 - r} ).
Exponential Functions
- Exponential growth is modeled by the equation ( A = P(1 + r)^t ), where ( A ) is the amount after time ( t ), ( P ) is the initial amount, and ( r ) is the growth rate.
- Exponential decay shares a similar formula: ( A = P(1 - r)^t ), demonstrating how quantities decrease over time.
- Compound interest formula is represented by ( A = P(1 + \frac{r}{n})^{nt} ), incorporating the compounding frequency ( n ).
Rates and Growth Factors
- In a situation where bacteria double every half hour, the growth rate is 100%, and the growth factor is 200% for that half-hour period; hence the hourly growth rate becomes 300%, leading to an hourly growth factor of 400%.
Identifying Exponential Behavior
- To determine if a table shows exponential behavior, check if there is a constant ratio ( \frac{A_n}{A_{n-1}} ) between consecutive terms, indicating a multiplicative relationship rather than a constant slope.
Properties of Exponents
- The rules of exponents, including ( a^x \cdot b^x = (ab)^x ), demonstrate how to simplify expressions involving multiplication of powers.
- Exponentiation also adheres to ( (b^n)^m = b^{n \cdot m} ) and ( a^x \cdot a^y = a^{x+y} ).
- For multiplying different bases raised to the same power, the rule is ( (ab)^x = (a^x)(b^x) ).
Additional Insights
- The geometric series can be utilized even when ( r > 1 ) to find the sum of a finite number of terms but should be approached with caution for contexts involving infinite series.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your knowledge on exponential and logarithmic functions with these flashcards. Each card provides a key term and its definition to help reinforce your understanding. Ideal for students looking to master the concepts of growth, decay, and geometric sequences.
