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Infinities 2 sequences and series

Infinities 2 sequences and series. 9:30 - 11:00 Geometric Sequences 11:30 - 13:00 Sequences, Infinity and ICT 14:00 - 15:30 Quadratic Sequences. Starter activity. Can you make your calculator display the following sequences? Find the 20 th term for each of these sequences.

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Infinities 2 sequences and series

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  1. Infinities 2sequences and series

  2. 9:30 - 11:00 Geometric Sequences 11:30 - 13:00 Sequences, Infinity and ICT 14:00 - 15:30 Quadratic Sequences

  3. Starter activity Can you make your calculator display the following sequences? Find the 20th term for each of these sequences.

  4. Geometric Sequences Can you find the next two terms of the following sequence 4, 8, 16, 32, ....?

  5. A sequence is geometric if Geometric sequences 4, 8, 16, 32, .... x2 x2 x2 x2 x2 where r is a constant called the common ratio

  6. Geometric sequences • orgeometric progressions, hence the GP notation • Different ways to describe this sequence: • By listing its first few terms: 4, 8, 16, 32, ... • By specifying the first term and the common ratio: • 1st term is 4 and common ratio is 2 or • By giving its nth term ? • By graphical representation ?

  7. Finding the nth term 4x1 4x2 4x4 4x8 4x16 4x20 4x21 4x22 4x23 4x24 nth term = 4x2n-1

  8. 4, 8, 16, ... is a divergent sequence

  9. Geometric sequences Can you find the next two terms of the following sequence? 0.2, 0.02, 0.002, .... • Can you describe this sequence in different ways? • By listing its terms: • By specifying the first term and the common ratio: • By finding its nth term: • By graphical representation:

  10. 0.2, 0.02, 0.002, ... is a convergent sequence The sequence converges to a certain value (or a limit number)

  11. e.g.it approaches 0 Another example of a convergent sequence: This convergent sequence also oscillates.

  12. Geometric sequences • 1. Can you generate (or find) the first 5 terms of the following GPs? • Seq A: • Seq B: • 2. Can you write down the nth term of these sequences? • 3. Are these sequences convergent or divergent? • Can you use the limit notation in your answers?

  13. Geometric sequences 1. What is the ratio and the 7th term for each of the following GPs? Seq A: 2, 10, 50, 250, ...? Seq B: 24,12, 6, 3, ....? Seq C: -27, 9, -3, 1, ....? Challenge 1 What if you want to find the 50th term of each of these sequences? How would you change your approach? Challenge 2 The 3rd term in a geometric sequence is 36 and the 6th term is 972. What is the value of the 1st term and the common ratio? Challenge 3 Q6 handout

  14. Geometric Series Suppose we have a 2 metre length of string . . . . . . which we cut in half We leave one half alone and cut the 2nd in half again . . . and again cut the last piece in half

  15. Continuing to cut the end piece in half, we would have in total In theory, we could continue for ever, but the total length would still be 2 metres, so This is an example of an infinite series.

  16. or is the Greek capital letter S, used for Sum

  17. Geometric series The sum of all the terms of a geometric sequence is called a geometric series. We can write the sum of the first n terms of a geometric series as: Sn = a + ar + ar2 + ar3 + … + arn–1 For example, the sum of the first 5 terms of the geometric series with first term 2 and common ratio 3 is: S5 = 2 + (2 × 3) + (2 × 32) + (2 × 33) + (2 × 34) = 2 + 6 + 18 + 54 + 162 = 242 When n is large, how efficient is this method?

  18. The sum of a geometric series Challenge: Can you follow the proof of the formula for the sum of the first n terms of a GS? (in pairs) Start by writing the sum of the first n terms of a general geometric series with first term a and common ratio r as: Multiplying both sides by r gives: Sn= a + ar + ar2 + ar3 + … + arn–1 Now if we subtract the first equation from the second we have: rSn= ar + ar2 + ar3 + … + arn–1 + arn rSn– Sn= arn – a Sn(r – 1) = a(rn – 1)

  19. Geometric series • Find the sum of the first 7 terms of the following GP: • 4, - 2, 1, . . . giving your answer correct to 3 significant figures. • Calculate: • Challenge • Is ? • What is as an exact fraction?

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