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Geometric Sequences and Series Part III

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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

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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

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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:

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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 dont need to learn the proof.

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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.

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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.

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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.

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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.

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

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

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

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

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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 dont 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.

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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 !!

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

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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

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Geometric Sequences and Series Sum to Infinity IF |r|