Post on 23-Aug-2020
Sequences and series part 1
1
Objective Deadlines / Progress R
ecurr
ence
rel
atio
ns
Find the next terms in a recurrence relation. Find a
formula for a recurrence relation.
Solve algebraic problems with recurrence relations e.g.
π’π+1 = 3π’π β π, Given that π’1 + π’2 + π’3 = 0 find the value of c
Understand sigma notation for series. Write a series
given the expression in sigma notation. Find a sum of
series from a problem in sigma notation.
Ari
thm
etic
ser
ies
Be able to prove the formula for the sum of an
arithmetic sequence.
Understand the structure
a, a+d β¦ and the last term as a+(n-1)d
Find the nth term of sequence
Find the sum of n terms
Solve algebraic problems such as when given the
second and fourth term find the sum to ten terms
Solve real life problems such as WB22
Solve problems where sigma notation is used
Geo
met
ric
ser
ies
Be able to prove the formula for the sum of an
geometric sequence.
Understand the structure
a, ar β¦ and the last term as πππβ1
Find the nth term of geometric sequence
Find the sum of n terms
Know when a geometric series ahs a sum to infinity
(when |π| < 1) and find the sum to infinity
Solve algebraic problems such as when given the
second and third term find the sum to ten terms
Solve problems where sigma notation is used
Including where the sequence does not start at the first
term
Sequences and series part 1
2
WB1 Work out the first 5 terms of each sequence
a) Un+1 = Un + 3, U1 = 3
b) Un+1 = 5Un β 2, U1 = 4
c) ππ = β(ππβ1)2 β 3, U1 = 4
d) ππ+1 = ππ
3+ππ, U1 = 2
Sequences and series part 1
3
WB2 Explore these sequences
i) Try changing the coefficients
ii) Try changing the initial value
a) tn+1 = 2tn β 7, t1 = 4
b) tn+1 = 3tn β 2, t1 = 1
c) tn+1 = 1 +2
π‘πβ1, t1 = 2
Make up your own recurrence formula
Try some different starting values
- Fractions
- Negative numbers
- Surds
Can you design a recurrence relation that
- is Constant
- is Oscillating
- Converges
Sequences and series part 1
4
WB3 Find a recurrence relation of the form ππ+1 = π΄ππ + π΅
For each of the following
a) 20, 17, 14, 11, 8, β¦.
b) 0.08, 0.4, 2, 10, 50, β¦.
c) 40, 48, 57.6, 69.12, β¦.
d) 4, 7, 16, 43, 124, β¦.
WB4 Find a recurrence relation for each of the following
a) 3, -7, 13, -27, -53, β¦.
b) 6, 12, 36, 164, 820, β¦.
c) 2, -4, 16, -256, β¦.
d) 1, 1
2 ,
1
3,
1
4,
1
5, β¦.
Sequences and series part 1
5
WB5 a sequence is defined by π’π+1 = 3π’π β π, and π’1 = 4 where π is a constant
a) Find an expression for π’2 in terms of c
b) Given that π’1 + π’2 + π’3 = 0 find the value of c
WB6 The nth term of a sequence is π’π, the sequence is defined by π’π+1 = ππ’π + π, where π & π are constants The first three terms of the sequence are Find π’1 = 2, π’2 = 5 and π’3 = 14
a) Show that π = β1 and find the value of π b) Find the value of π’4
Sequences and series part 1
6
WB7 The nth term of a sequence is π’π The sequence is defined by π’π+1 = ππ’π + π , where
p and q are constants
The first three terms are π’1 = β10 π’2 = 11 π’3 = 0.5
a) Show that π = 6 and find the value of p
b) find the value of π’4
The limit of π’π as n tends to infinity is L
c) Write an equation for L and solve it to find the value of L
WB8 a sequence is defined by π’π+1 = 2π’π + 3, and π’1 = π where π is a positive integer a) Write an expression for π’2 in terms of k b) Show that π’3 = 4π + 9 c) Find π’1 + π’2 + π’3 + π’4 in terms of k, in its simplest form and show that the result is divisible by 3
Sequences and series part 1
7
SIGMA notation
WB9 Write the series
a) β (3π + 5)ππ=1
b) β (3π + 5)10π=1
c) β (3π + 5)20π=8
d) β (3π2 + 1)6π=1
Sequences and series part 1
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WB10 Write these series in sigma notation
a) 3 + 7 + 11 + 15 + 19 + β― β¦ + 79
b) 1 + 4 + 9 + 16 + 25 + β― β¦ + 10000
c) 76 + 69 + 62 + 55 + 48 + β― β¦ β 92
d) 2 + 16 + 54 + 128 + 250 + 432 + 686
WB11 Evaluate:
a) β (π2 + 1)8π=5
b) β (2π)5π=1
c) β (β1)πβ18
π=1 (3π β 1)
d) β 10π=4
π
12βπ
Sequences and series part 1
9
Sequences and series part 1
10
Arithmetic sequences and Series
An arithmetic sequence is a sequence in which, beginning with the second term, each term is found by adding the same value (d) to the previous term. The number d is called the common difference. What will be the nth term of these arithmetic sequences? 4Β½, 6, 7Β½, 9, 10Β½, β¦
11, 22, 33, 44, 55, β¦
0.4, 0.6, 0.8, 1.0, 1.2, β¦
14, 36, 58, 80, 102, β¦
-4, 10, 24, 38, 52, β¦
work out the 20th term of each sequence.
WB12 Once upon a time in a maths lesson a bright student called Johann Gauss, who was always
completing work before everyone else asked his teacher for something else to do. The
teacher said βgo and add up the numbers from 1 to 100.β thinking this would keep the
Johann busy.
But he came back a minute later with the answer. How did he work it out?
Gauss was born 1777 in a small German city. The son of peasant parents (both were illiterate).
Sequences and series part 1
11
WB13 Structure of an arithmetic sequence: Structure of an arithmetic series:
π, π + π, π + 2π, π + 3π, β¦ β¦ . π + (π β 2)π, π + (π β 1)π
An arithmetic sequence has first term, a, common difference, d, number of terms, n Find the nth term of a sequence using π + (π β 1)π Find the difference d by subtracting two consecutive terms
a) Write the values of a, d and the 200th term of the sequence of odd numbers 1, 3, 5, 7 β¦
The sum of n terms of a sequence is given by a Series:
ππ = β(π + (π β 1)π) =
π
π=1
π + (π + π ) + (π + ππ ) + β― β¦ + [π + (π β π)π ]
π) β(ππ + ππ) =
π
π=π
Sequences and series part 1
12
WB14 Find the number of terms in each of these arithmetic series
ππ + ππ + ππ + ππ + β¦ β¦ + ππ
ππ + ππ + ππ + ππ + β¦ β¦ + (βππ)
ππ
π+ π +
ππ
π+
ππ
π+ β¦ β¦ + ππ
WB15 The 5th term of an arithmetic sequence is 24 and the 9th term is 4
a) Find the first term and the common difference
b) The last term of the sequence is -36
How many terms are in this sequence?
Sequences and series part 1
13
WB16 Sum of terms formula: version 1
The sum of terms is given by adding the first and last terms, a and L, then multiplying by
the number of terms, n, then dividing by 2.
Write a formula for this
Show that your formula works!
WB16 Sum of terms formula: version 2
What will be an alternative formula if we substitute the last term
as πΏ = π + (π β 1)π
Show that your formula works!
Sequences and series part 1
14
WB17 The third term of an arithmetic sequence is 11 and the seventh term is 23.
Find the first term and the common difference.
If the rth term is 62 then find r and find the sum of r terms
WB18 An arithmetic series has first term 6 and common difference 2Β½ . Find the least value of n
for which the nth term exceeds 1000
Sequences and series part 1
15
You should be able to derive (give a proof) of the formula ππ =π
2[2π + (π β 1)π]
in about six or seven clear steps with correct notation: LHS and RHS; and
short annotations to explain the steps
Sequences and series part 1
16
WB19 The first term of an arithmetic sequence is 3, the fourth term is -9. What is the sum of the
first 24 terms?
WB20 The first term of an arithmetic sequence is 2, the sum of the first 10 terms is 335. Find the
common difference
Sequences and series part 1
17
WB21 An arithmetic sequence for building each step of a spiral has first two terms 7.5 cm
and 9 cm. What will be (i) the length of the 40th line of the spiral
(ii) the total length of the spiral after 40 steps?
WB22 Simultaneous equations
An arithmetic sequence is used for modelling population growth of a Squirrel colony
starting at three thousand in the year 2000. The 2nd and 5th numbers in the sequence are 14
and 23 showing the increase in population those years. Find: (i) the first increase in
population (ii) the 16th increase (iii) the population after 16 years?
Sequences and series part 1
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WB23 Simultaneous equations
The first three terms of an arithmetic sequence are (4x β 5), 3x and (x + 13) respectively
a) Find the value of x b) Find the 23rd term
WB24 The sum of the first ten terms of an arithmetic sequence is 113. The first term is a and the
common difference is d.
a) Show that 10π + 45π = 113
b) Given the sixth term is 12, write a second equation in a and d
c) Find the values of a and d
Sequences and series part 1
19
WB25 The 11th term of an arithmetic sequence is equal to 2 times the 4th term. The 19th term is 44
a) Find the first term and the common difference
b) Find the sum of the first 60 terms
WB26 Quadratic!
The sum of an arithmetic sequence to n terms is 450
The 2nd and 4th terms are 40 and 36. Find the possible values of n
Sequences and series part 1
20
Using Sigma notation
WB27 Evaluate
a) β (7π β 3)461
b) β (3π + 5)221
c) β (4 + 10π)211
WB28 Find the value of n such that :
a) β (3π β 7) = 217ππ=1
b) β (65 β 4π) = β882ππ=1
Sequences and series part 1
21
WB29 Evaluate these: (π€βπππ π β 1)
π) β(3π)
30
π=5
π) β (3π β 5)
40
π=10
π) β (2π + 6)
38
π=22
WB30 ππππ ππππ β(3π + 4)
π
1
= 3 β π
π
1
+ 4π
Sequences and series part 1
22
WB31
ππ£πππ’ππ‘π β(2π)
25
1
Evaluate these using the previous result
β(2π + 3)
25
1
β(4π)
25
1
β(4π β 7)
25
1
WB32 the nth term of an arithmetic sequence is (6π β 8)
a) Write down the first three terms of this sequence
b) State the value of the common difference
c) Show that β (6π β 8) = π(3π β 5)ππ=1
Sequences and series part 1
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WB33 A sequence of terms π’1, π’2, π’3 β¦ . . Is defined by π’π = 20 β4
5 π
a) Write down the exact values of the value of π’1, π’2, and π’3 b) find the value k such that of π’π = 0
c) Find β π’π19π=1
WB34 a) In the arithmetic series π + 2π + 3π + β― + 120 k is a positive integer and a factor of 120.
Find an expression for the number of terms in this series
b) Show that the sum of the series in (b) is 60 +7200
π
c) Find, in terms of k, the 60th term of the arithmetic sequence
(2k + 2), (4π + 4), (6π + 6), β¦ giving your answer in simplest form