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C2: Geometric Series

C2: Geometric Series. Dr J Frost (jfrost@tiffin.kingston.sch.uk) . Last modified: 24 th September 2013. Types of series. common difference . ?. +3. +3. +3. This is a:. 2, 5, 8, 11, 14, …. ?. Arithmetic Series. common ratio . ?. 3, 6, 12, 24, 48, …. ?. Geometric Series.

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C2: Geometric Series

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  1. C2: Geometric Series Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 24th September 2013

  2. Types of series common difference ? +3 +3 +3 This is a: 2, 5, 8, 11, 14, … ? Arithmetic Series common ratio ? 3, 6, 12, 24, 48, … ? Geometric Series

  3. Common Ratio Identify the common ratio : ? 1 2 ? ? 3 ? 4 ? 5 ? 6 ? 7

  4. Common Ratio Exam Question May 2013 (Retracted) Hint for (a): the common ratio between the first and second terms, and the second and third terms, is the same. a ? b ?

  5. th term Arithmetic Series Geometric Series ? ? Determine the following: 3, 6, 12, 24, … ? ? 40, -20, 10, -5, … ?

  6. Another Common Ratio Example The numbers and form the first three terms of a positive geometric sequence. Find: a) The possible values of . b) The 10th term in the sequence. But there are no negative terms so ?

  7. Missing information The second term of a geometric sequence is 4 and the 4th term is 8. The common ratio is positive. Find the exact values of: The common ratio. The first term. The 10th term. Dividing (1) by (2) gives Substituting, ? Bro Tip: Explicitly writing first helps you avoid confusing the th term with the ‘sum of the first terms’ (the latter of which we’ll get onto).

  8. th term with inequalities What is the first term in the geometric progression to exceed 1 million? ?

  9. Exam Question Edexcel June 2010 ? ? ? ? ?

  10. Sum of the first terms Arithmetic Series Geometric Series ? ? Technically you could be asked in an exam the proof of the sum of a geometric series (it once came up!) So let’s prove it…

  11. Sum of the first terms Geometric Series Find the sum of the first 10 terms. ? ? ? ? ? ? ? ?

  12. Summation Notation Find ? ? ? ?

  13. Harder Questions: Type 1 Find the least value of such that the sum of to terms would exceed 2 000 000. ? An investor invests £2000 on January 1st every year in a savings account that guarantees him 4% per annum for life. If interest is calculated on the 31st of December each year, how much will be in the account at the end of the 10th year? ?

  14. Exercise 7D Find the sum of the following geometric series (to 3dp if necessary). a) (8 terms) c) (6 terms) e) h) The sum of the first three terms of a geometric series is 30.5. If the first term is 8, find the possible values of . Jane invest £4000 at the start of every year. She negotiates a rate of interest of 4% per annum, which is paid at the end of the year. How much is her investment worth at the end of (a) the 10th year and (b) the 20th year. (a) (b) A ball is dropped from a height of 10m. It bounces to a height of 7m and continues to bounce. Subsequent heights to which it bounces follow a geometric sequence. Find out: How high it will bounce after the fourth bounce, The total distance travelled until it hits the ground for a sixth time. Find the least value of such that the sum to terms would first exceed 1.5 million. 1 ? ? ? ? 2 ? 4 ? ? 5 ? ? 6 ?

  15. Different types of series What can you say about the sum of each series up to infinity? ? 1 + 2 + 4 + 8 + 16 + ... This is divergent – the sum of the values tends towards infinity. This is divergent – the sum of the values tends towards infinity. But arguably, the sum of the natural numbers is . ? 1 + 2 + 3 + 4 + 5 + ... 1 + 0.5 + 0.25 + 0.125 + ... This is convergent – the sum of the values tends towards a fixed value, in this case 2. ? Just for fun... ? This is divergent. This is known as the Harmonic Series This is convergent . This is known as the Basel Problem, and the value is . ?

  16. Sum to Infinity Think about our formula for the sum of the first terms. If we make infinity, what do we require of for not to be infinity (i.e. we want to keep the series convergent). And what will the formula become? ? ? Restriction on :

  17. Examples ? ? ? ? ? ? ? ? ? ? ? ?

  18. A somewhat esoteric Futurama joke explained Bender (the robot) manages to self-clone himself, where some excess is required to produce the duplicates (e.g. alcohol), but the duplicates are smaller versions of himself. These smaller clones also have the capacity to clone themselves. The Professor is worried that the total amount mass consumed by the growing population is divergent, and hence they’ll consume to Earth’s entire resources.

  19. A somewhat esoteric Futurama joke explained This simplifies to The sum is known as the harmonic series, which is divergent.

  20. Another Example The sum to 4 terms of a geometric series is 15 and the sum to infinity is 16. a) Find the possible values of . b) Given that the terms are all positive, find the first term in the series. ? ?

  21. Another Example Edexcel May 2011 ? ? ? ?

  22. Exercises Exercise 7D Q6, 7 Exercise 7E Q8 Exercise 7F Q10

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