1. Geometric Sequences and Series Part III

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

3. Geometric Sequences and Series A geometric sequence or geometric progression (G.P.) is of the form The n th term of an G.P. is

4. Geometric Sequences and Series Exercises 1. Use the formula for the n th term to find the term indicated of the following geometric sequences (b) (c) (a) Ans:

5. Geometric Sequences and Series e.g.1 Evaluate Writing out the terms helps us to recognize the G.P. Summing terms of a 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. Geometric Sequences and Series Subtracting the expressions gives With 5 terms of the general G.P., we have TRICK Multiply by r: Move the lower row 1 place to the right Summing terms of a G.P.

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

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

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

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

11. Geometric Sequences and Series For our series Using Summing terms of a G.P.

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

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

14. Geometric Sequences and Series 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. Solution: As there are so few terms, we don’t need the formula for a sum 3 rd term + 4 th term = 4 ( 1 st term + 2 nd term ) Divide by a since the 1 st term, a, cannot be zero: Summing terms of a G.P.

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

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

17. Geometric Sequences and Series SUMMARY  A geometric sequence or geometric progression (G.P.) is of the form  The n th term of an G.P. is  The sum of n terms is or

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

19. Geometric Sequences and Series 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 the G.P. 4  2 + 1 +... giving your answer correct to 3 significant figures.

20. Geometric Sequences and Series Exercises 1. Solution: 2 + 8 + 32 +... 2. Solution: 4  2 + 1 +... ( 3 s.f. )