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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.

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Geometric Sequences and Series Part III

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  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

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