GEOMETRIC SERIES Feb 11 th 2015. Geometric Series is called the partial sum.
Geometric series
Transcript of Geometric series
32: Geometric 32: Geometric Sequences and SeriesSequences and Series
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© Christine Crisp
““Teach A Level Maths”Teach A Level Maths”
Vol. 1: AS Core Vol. 1: AS Core ModulesModules
Geometric Sequences and Series
Module C2
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Geometric Sequences and Series
There is a legend that Zarathustra, a Persian, invented chess to give interest to the life of the king who was bored. For his reward, Zarathustra asked for a quantity of grain, according to the following rules.
Geometric Sequences and Series
2 on the next,
1 grain was to be placed on the 1st square of the chess board,
4 on the 3rd
and so on, doubling the number each square.
How many must be placed on the 64th square?
Geometric Sequences and Series
...,8,4,2,1
We have a sequence:
Each term is twice the previous term, so by the 64th term we have multiplied by 2 sixty-three timesWe have 63
64 2u
approximately or 9 followed by 18 zeros!
18109
Geometric Sequence
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.
Although with a calculator we can see that the sum is 186, 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
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.
To get this version of the formula, we’ve multiplied the 1st form by
1
1
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 e.g. 2
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
Using the factor theorem:
0))(1( r 2r
factor a is )1(04411)1( rf
4
We need to solve the cubic equation
04423 rrr
The squared term in is 2r04423 rrr
As we already have , the quadratic factor is complete. The middle term is zero.
2r
Summing terms of a G.P.
Geometric Sequences and Series
factor anot is )1(04411)1( rf
Using the factor theorem:
0))(1( r 2r
factor a is )1(04411)1( rf
4
We need to solve the cubic equation
04423 rrr
0)2)(2)(1( rrrSince we were told we
get 1r 2r
42 rFactorizing
Summing terms of a G.P.
Geometric Sequences and Series
e.g. 4 £100 is invested every year on the first of January and earns compound interest at the rate of 4% per annum. Find the amount by the end of the 5th year, to the nearest penny.Solution: The last £100 is invested for 1 year
only.
041100 The 4th £100 is invested for 2 years so at the end is worth
2041100
At the end, this £100 is worth
100040100 )0401(100
100 is a common factor
Summing terms of a G.P.
Geometric Sequences and Series
1
)1(
r
raS
n
n
1041
1041104 5
5
S
305635 £S (nearest penny)
At the end of the 5 years, the total invested will be worth
52 041100...041100041100
This is a G.P. with 104041100 a
5,041 nr
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
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. )
Geometric Sequences and Series