Questions and Answers
In a geometric sequence, what is the name for the constant multiplied by each term to find the next term?
common ratio
What is the name of a geometric sequence whose terms change between positive and negative?
alternating (sequence)
What type of geometric series never reaches an end?
infinite
Is the sequence 64, 16, 4, 1,... converging or diverging?
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Is the sequence 1/4, 1/2, 1, 2,... converging or diverging?
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Is the sequence 7, -14, 28, -56,... converging or diverging?
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Find the next term in the following geometric sequence: 3, -12, 48,...
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Find the fourth term of a geometric sequence if a = 6 and r = 2. Use the formula aₙ = aⁿ⁻¹.
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Find the sum of the first 2 terms of a geometric sequence if a = 2 and r = 4. Use the formula sₙ = ₐ(1-rⁿ)/1 - r.
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A patient receives 10 mL of medicine every day at the same time. If 50% of the medicine metabolizes during the day, how much medicine will be left in his body after 3 days?
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Study Notes
Geometric Sequences and Series
- Common Ratio: The constant used to multiply each term in a geometric sequence to obtain the next term.
- Alternating Sequence: A geometric sequence where the terms switch between positive and negative values.
- Infinite Series: A type of geometric series that continues indefinitely without reaching an end.
Convergence and Divergence
- Converging Series: A series where the terms approach a specific value, as exemplified by the sequence 64, 16, 4, 1,....
- Diverging Series: A series where the terms do not approach a specific limit. Examples include sequences such as 1/4, 1/2, 1, 2,... and 7, -14, 28, -56,...
Calculations in Geometric Sequences
- Next Term Calculation: For the geometric sequence 3, -12, 48, the next term is -192.
- Fourth Term Calculation: In a geometric sequence where the first term (a) is 6 and the common ratio (r) is 2, the fourth term is 48, calculated using the formula aₙ = aⁿ⁻¹.
- Sum of Terms Calculation: The sum of the first two terms of a geometric sequence with a = 2 and r = 4 is calculated to be 10, using the formula sₙ = ₐ(1-rⁿ)/(1 - r).
Practical Application
- Medicine Metabolism Problem: A patient receiving 10 mL of medicine daily, with 50% metabolized each day, will have 8.75 mL of medicine remaining in their body after 3 days.
Studying That Suits You
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Description
Test your knowledge of geometric sequences with these flashcards from Abeka's Algebra II curriculum for the 2022-2023 year. Focus on terms like common ratio and alternating sequences to reinforce your understanding and retention of key concepts.
