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

[table][br][tr][br] [td][b]Factual Inquiry Questions[/b][br] [list][br] [*]What defines a geometric sequence, and how is the nth term of a geometric sequence calculated?[br] [*]How is the sum of the first n terms of a geometric series determined?[br] [/list][br] [/td][br] [td][b]Conceptual Inquiry Questions[/b][br] [list][br] [*]Why does a geometric series converge or diverge, and what criteria determine this behavior?[br] [*]How can the concept of geometric sequences and series be applied to model exponential growth or decay in real-world scenarios?[br] [/list][br] [/td][br] [td][b]Debatable Inquiry Questions[/b][br] [list][br] [*]Is the study of geometric sequences and series more relevant to theoretical mathematics or to practical applications in fields such as finance and physics?[br] [*]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?[br] [*]How might advancements in technology and computational methods impact the application and significance of geometric sequences and series in solving real-world problems?[br] [/list][br] [/td][br][/tr][br][/table][br]

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

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. [br]Geometric sequences 1 [url=https://youtu.be/JwNdNSgq4uI]https://youtu.be/JwNdNSgq4uI[/url]  Geometric sequences 2  [url=https://youtu.be/_f1k58Hyy1c]https://youtu.be/_f1k58Hyy1c[/url]  Geometric sequences - Harder [url=https://youtu.be/2uEVctIHihg]https://youtu.be/2uEVctIHihg[/url]  Geometric sequences - Sum to infinity [url=https://youtu.be/Jpm1F9ifMRg]https://youtu.be/Jpm1F9ifMRg[/url]  Geometric sequences - Sum to infinity past paper question [url=https://youtu.be/YAR_epDxhec]https://youtu.be/YAR_epDxhec[/url]  Sigma notation [url=https://youtu.be/tIkJNLWAbJs]https://youtu.be/tIkJNLWAbJs[/url] 

Testing understanding with exam style questions