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Geometric (AA/AI SL 1.3)

Keywords

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Geometric Sequence等比数列등비 수열等比数列
nth Termn番目の項n번째 항第n项
Geometric Series等比級数등비 급수等比级数
Sum of the First n Terms最初のn項の和첫 n 항의 합前n项和
Convergence収束수렴收敛
Divergence発散발산发散
Exponential Growth指数関数的成長지수 성장指数增长
Exponential Decay指数関数的減少지수 감소指数衰减
Common Ratio公比공비公比
Sigma Notationシグマ記法시그마 표기법Σ表示法
Sum to Infinity無限大への和무한대까지의 합无穷级数的和
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Inquiry questions

Factual Inquiry Questions
  • What defines a geometric sequence, and how is the nth term of a geometric sequence calculated?
  • How is the sum of the first n terms of a geometric series determined?
Conceptual Inquiry Questions
  • Why does a geometric series converge or diverge, and what criteria determine this behavior?
  • How can the concept of geometric sequences and series be applied to model exponential growth or decay in real-world scenarios?
Debatable Inquiry Questions
  • Is the study of geometric sequences and series more relevant to theoretical mathematics or to practical applications in fields such as finance and physics?
  • Can the principles of geometric sequences be effectively used to understand complex phenomena in nature, such as population dynamics or the spread of diseases? How?
  • How might advancements in technology and computational methods impact the application and significance of geometric sequences and series in solving real-world problems?

The Great Geometric Quest

Exploration Title: The Great Geometric Quest Objective: Embark on a quest to unlock the secrets of geometric sequences and series. Use the given applet to uncover the hidden patterns and sum up your findings! Mission Steps: 1. Sequence Sleuth: - Find the 10th term in the sequence with a first term of 3 and a common ratio of 2. - Predict what the 100th term might be and discuss the practicality of its magnitude. 2. Sum Search: - Calculate the sum of the first 10 terms of the geometric sequence from step 1. - Hypothesize about the sum of the first 100 terms. How does it compare to the 100th term alone? 3. Ratio Riddle: - For the geometric sequence 5, 10, 20, ..., deduce the common ratio without using the applet. - Calculate the 15th term and the sum of the first 15 terms manually, then check with the applet. Questions for Investigation: 1. Real-life Relevance: - How can understanding geometric sequences help in real-life scenarios like finance (interest rates) or computer science (data storage)? 2. Series Saturation: - Is there a limit to the sum of a geometric series? Experiment with different ratios (less than 1, equal to 1, greater than 1) to find out. 3. Pattern Pursuit: - Can you create a geometric sequence that alternates between positive and negative without changing the common ratio? Engagement Activities: - "Sequence Scramble": Given the 5th and 9th terms of a geometric sequence, work backward to find the first term and common ratio. - "Sum Sprint": Challenge each other to find the sum of the first 'n' terms of a given sequence as fast as possible. Through this investigation, become the master of multiplication patterns and understand how simple ratios build complex sequences and staggering sums.

Part 2 - Checking understanding

Watch through these two videos before attempting the questions
Alternatively, if you feel confident with the theory, watch these brief videos with worked examples of exam-style questions. Geometric sequences 1 https://youtu.be/JwNdNSgq4uI  Geometric sequences 2  https://youtu.be/_f1k58Hyy1c  Geometric sequences - Harder https://youtu.be/2uEVctIHihg  Geometric sequences - Sum to infinity https://youtu.be/Jpm1F9ifMRg  Geometric sequences - Sum to infinity past paper question https://youtu.be/YAR_epDxhec  Sigma notation https://youtu.be/tIkJNLWAbJs 

If the first term of a geometric sequence is 3 and the common ratio is 4, what is the 4th term?

Select all that apply
  • A
  • B
  • C
  • D
Check my answer (3)

If the first term is 95, and the third term is 380. Find the sum to 5th term.

Part 3 - Testing understanding with exam style questions

Testing understanding with exam style questions

[MAA 1.4] GEOMETRIC SEQUENCES

[MAA 1.4] GEOMETRIC SEQUENCES_solutions

Lesson Plan- Understanding Geometric Sequences and Series

Geometric- Intuition pump (thought experiments and analogies)