Geometric Sequences and Series Part III
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Transcript of Geometric Sequences and Series Part III
Geometric Sequences Geometric Sequences and Seriesand Series
Part IIIPart III
Geometric Sequences and Series
632...,8,4,2,1
The sequence
is an example of aGeometric sequence
A sequence is geometric if
rterm previous
term each
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
1 nn aru
...,,,, 32 ararara
Geometric Sequences and Series
Exercises1. Use the formula for the nth term to find the
term indicated of the following geometric sequences
term th6...,32,8,2
term th5...,4
3,3,12
term th7...,0020,020,2.0
(b)
(c)
(a)
Ans: 2048)4(2 5
Ans: 64
3
4
112
4
Ans: 00000020)1.0(20 6 .
Geometric Sequences and Series
e.g.1 Evaluate
Writing out the terms helps us to recognize the G.P.
5
1
)2(3n
n
5432 )2(3)2(3)2(3)2(3)2(3
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.
Geometric Sequences and Series
4325 ararararaS
Subtracting the expressions gives
With 5 terms of the general G.P., we have
TRICK Multiply by r: 5432
5 arararararrS
Move the lower row 1 place to the right
43255 arararararSS
5432 ararararar
Summing terms of a G.P.
Geometric Sequences and Series
Subtracting the expressions gives
With 5 terms of the general G.P., we have
Multiply by r:
and subtract
54325 arararararrS
5432 ararararar
43255 arararararSS
4325 ararararaS
Summing terms of a G.P.
Geometric Sequences and Series
5432 ararararar
Subtracting the expressions gives
With 5 terms of the general G.P., we have
Multiply by r:
555 ararSS
4325 ararararaS
54325 arararararrS
43255 arararararSS
Summing terms of a G.P.
Geometric Sequences and Series
r
raS
1
)1( 5
5
r
raS
n
n
1
)1(
Similarly, for n terms we
get
555 ararSS So,
Take out the common factors
and divide by ( 1 – r )
)1()1( 5rr aS5
Summing terms of a G.P.
Geometric Sequences and Series
gives a negative denominator if r
> 1
r
raS
n
n
1
)1(The formula
1
)1(
r
raS
n
n
Instead, we can use
Summing terms of a G.P.
Geometric Sequences and Series
5432 )2(3)2(3)2(3)2(3)2(3 For our series
12
)12(6 5
nS
1
)31(6
186
52,6 nra and
1
)1(
r
raS
n
nUsing
Summing terms of a G.P.
Geometric Sequences and Series
Find the sum of the first 20 terms of the geometric series, leaving your answer in index form
31
312 20
20
Sr
raS
n
n
1
)1(
...541862 EX
2
6,2
raSolution
:
3
1
3
We’ll simplify this answer without using a calculator
Summing terms of a G.P.
Geometric Sequences and Series
4
312 20
2
31 20
There are 20 minus signs here and 1 more outside the bracket!
31
312 20
20
S
1
2
Summing terms of a G.P.
Geometric Sequences and Series
e.g. 3In 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 3rd term + 4th term = 4( 1st term + 2nd
term ))(432 araarar
Divide by a since the 1st term, a, cannot be zero:
)1(432 rrr 04423 rrr
Summing terms of a G.P.
Geometric Sequences and Series
factor anot is )1(04411)1( rf
Should use the factor theorem:
factor a is )1(04411)1( rf
We need to solve the cubic equation
04423 rrr
Summing terms of a G.P.
We will do this soon !!
factor a is )2(04848)2( rf
factor a is )2(04848)2( rf
factors the are )2)(2)(1( rrr
Geometric Sequences and Series
The solution to this cubic equation is therefore
04423 rrr
0)2)(2)(1( rrr
Since we were told we get
1r 2r
Summing terms of a G.P.
Geometric Sequences and Series
SUMMARY
r
raS
n
n
1
)1(
A geometric sequence or geometric progression (G.P.) is of the form
The nth term of an G.P. is
1 nn aru
...,,,, 32 ararara
The sum of n terms is
1
)1(
r
raS
n
no
r
Geometric Sequences and Series
Sum to Infinity
IF |r|<1 then
)(1
))1(1(
r
aS
n
r
aS
1
Because (<1)∞ = 0
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
3
)14(2 15
15
S
1
)1(
r
raS
n
n15,4,2 nra
14
)14(2 15
15
S
1. Solution: 2 + 8 + 32 + . . .
501
5014 15
15
S
r
raS
n
n
1
)1(15,50,4 nra
2. Solution: 4 2 + 1 + . . .
67215 S( 3 s.f. )