# Geometric Sequences and Series Geometric Sequences and Series

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Geometric Sequences and Series

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

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

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

Geometric Sequences and Series 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.

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

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

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

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

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

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

Geometric Sequences and Series 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 Solution: We’ll simplify this answer without using a calculator

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

Geometric Sequences and Series Summing terms of a G. P. e. g. 3 In a geometric sequence, the sum of the 3 rd and 4 th terms is 4 times the sum of the 1 st and 2 nd 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:

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

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

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

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

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.

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