1 / 20

Geometric Sequences and Series Part III

Geometric Sequences and Series Part III. A sequence is geometric if. The sequence. is an example of a. Geometric sequence. where r is a constant called the common ratio. In the above sequence, r = 2. A geometric sequence or geometric progression (G.P.) is of the form.

blaine
Download Presentation

Geometric Sequences and Series Part III

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Geometric Sequences and Series Part III

  2. A sequence is geometric if The sequence is an example of a Geometric sequence where r is a constant called the common ratio In the above sequence, r = 2

  3. A geometric sequence or geometric progression (G.P.) is of the form The nth term of an G.P. is

  4. Ans: Ans: Ans: Exercises 1. Use the formula for the nth term to find the term indicated of the following geometric sequences (a) (b) (c)

  5. Summing terms of a G.P. e.g.1 Evaluate Writing out the terms helps us to recognize the G.P. With a calculator we can see that the sum is 186. But we need a formula that can be used for any G.P. The formula will be proved next but you don’t need to learn the proof.

  6. Move the lower row 1 place to the right Summing terms of a G.P. With 5 terms of the general G.P., we have TRICK Multiply by r: Subtracting the expressions gives

  7. and subtract Summing terms of a G.P. With 5 terms of the general G.P., we have Multiply by r: Subtracting the expressions gives

  8. Summing terms of a G.P. With 5 terms of the general G.P., we have Multiply by r: Subtracting the expressions gives

  9. So, Summing terms of a G.P. Take out the common factors and divide by ( 1 – r ) Similarly, for n terms we get

  10. The formula gives a negative denominator if r > 1 Instead, we can use Summing terms of a G.P.

  11. For our series Using Summing terms of a G.P.

  12. Solution: Summing terms of a G.P. EX Find the sum of the first 20 terms of the geometric series, leaving your answer in index form We’ll simplify this answer without using a calculator

  13. There are 20 minus signs here and 1 more outside the bracket! Summing terms of a G.P.

  14. e.g. 3 In a geometric sequence, the sum of the 3rd and 4th terms is 4 times the sum of the 1st and 2nd terms. Given that the common ratio is not –1, find its possible values. Summing terms of a G.P. Solution: As there are so few terms, we don’t need the formula for a sum 3rd term + 4th term = 4( 1st term + 2nd term ) Divide by a since the 1st term, a, cannot be zero:

  15. We need to solve the cubic equation Summing terms of a G.P. Should use the factor theorem: We will do this soon !!

  16. The solution to this cubic equation is therefore Since we were told we get Summing terms of a G.P.

  17. SUMMARY • A geometric sequence or geometric progression (G.P.) is of the form • The nth term of an G.P. is • The sum of n terms is or

  18. Sum to Infinity IF |r|<1 then 0 Because (<1)∞ = 0

  19. Exercises 1. Find the sum of the first 15 terms of the following G.P., giving the answers in index form 2 + 8 + 32 + . . . 2. Find the sum of the first 15 terms of theG.P. 4 - 2 + 1 + . . .giving your answer correct to 3 significant figures.

  20. 1. Solution: 2 + 8 + 32 + . . . 2. Solution: 4 - 2 + 1 + . . . ( 3 s.f. ) Exercises

More Related