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Exploring Sequences: Patterns, General Terms, and Future Terms Calculation

This educational resource delves into various sequences and their patterns, providing clear definitions and numerous examples to illustrate key concepts. Topics covered include the identification of terms, the definition of arithmetic sequences, and methods to establish general terms for different sequences. Students are encouraged to explore sequences such as 3, 6, 12, 24 and 1, 2, 4, 7, 11, as well as challenges involving term calculations and real-world applications. Discover the beauty of sequences in mathematics!

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Exploring Sequences: Patterns, General Terms, and Future Terms Calculation

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  1. Sequences Tuesday December 18, 2012

  2. WARM UP Describe the pattern and write the next 3 terms. a) 3, 6, 12, 24, ... b) 1, 2, 4, 7, 11, 16, ... c) 2, 5, 10, 17, 26, ...

  3. Definitions • Sequence – a set of numbers, separated by commas, arranged in an order. • Term - a number in a sequence. Subscripts are usually used to identify the position of the terms • First term: t1 • Second term: t2 • Third term: t3 • nth term: tn • A sequence may stop or continue indefinitely • Discrete Sequence: 5, 7, 9, 11 • Continuous Sequence: 80, 40, 20, 10, … (dots indicate that sequence is continuing indefinitely) t1 , t2 , t3 , …. tn

  4. Real world applications Archimedean Spiral t1 = 1 t2 = 2 t3 = 3 t4 = 4 Spirals Sea shells Cyclones Snail shells Finger tips Galaxy Etc. Exponential Spiral t1 = 1 t2 = 2 t3 = 3 t4 = 4

  5. General Term • A formula or rule, labelled tn, that expresses each term of a sequence as a function of its position, n.

  6. Example 1 a) Determine the general term for the sequence 3, 5, 7, 9, … • Use trial and error for now. • Write a General Term: tn = b) What would the 37th term be?

  7. Example 2 a) Determine the general term for the sequence 1, 3, 5, 7, 9, … • Use trial and error for now. • Write a General Term: tn = b) What would the 100th term be?

  8. Example 3 a) Determine the general term for the sequence 1, 3, 9, 27, … • Use trial and error for now. • Write a General Term: tn = b) What would the 55th term be?

  9. Example 4 a) Write the first 5 terms of the sequence tn = 3n - 2 b) Given the formula for the nth term, f(n) = 2n2 - 1 in function notation, find t8.

  10. Arithmetic Sequences

  11. Arithmetic Sequences • A sequence that has the same difference, the common difference, between any pair of consecutive terms. General Term: a- first term d- common difference n- term number

  12. Example 5 • Determine a formula that defines the arithmetic sequence 3, 12, 21, 30, … • Write a General Term: tn = b) What would the 27th term be?

  13. Example 6 What is the 33rd term of the sequence 18, 11, 4, -3, …

  14. Example 7 The 7th term of an arithmetic sequence is 53 and the 11th term is 97. Determine the 100th term.

  15. Example 8 Find the number of terms there are in the sequence: -3, 2, 7, …, 152

  16. Example 9 The rates of a mini grand-prix race are $7 for a license fee and $3 per lap. • What would the rate for 10 laps be? • How many laps can you do with $22?

  17. HOMEFUN!! • Worksheet – Sequences • Textbook 7.1 Pg. 424 – 425, #5 – 12, 15

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