Text of Geometric Sequences and Series Part III. Geometric Sequences and Series The sequence is an example...
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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|