sequences and series part 1 · sequences and series part 1 5 wb5 a sequence is defined by 𝑛+1=3...

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.

𝑢𝑛+1 = 3𝑢𝑛 − 𝑐, Given that 𝑢1 + 𝑢2 + 𝑢3 = 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 … 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 … and the last term as 𝑎𝑟𝑛−1

Find the nth term of geometric sequence

Find the sum of n terms

Know when a geometric series ahs a sum to infinity

(when |𝑟| < 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


2

WB1 Work out the first 5 terms of each sequence

a) Un+1 = Un + 3, U1 = 3

b) Un+1 = 5Un – 2, U1 = 4

c) 𝑈𝑛 = √(𝑈𝑛−1)2 − 3, U1 = 4

d) 𝑈𝑛+1 = 𝑈𝑛

3+𝑈𝑛, U1 = 2


3

WB2 Explore these sequences

i) Try changing the coefficients

ii) Try changing the initial value

a) tn+1 = 2tn – 7, t1 = 4

b) tn+1 = 3tn – 2, t1 = 1

c) tn+1 = 1 +2

𝑡𝑛−1, 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


4

WB3 Find a recurrence relation of the form 𝑈𝑛+1 = 𝐴𝑈𝑛 + 𝐵

For each of the following

a) 20, 17, 14, 11, 8, ….

b) 0.08, 0.4, 2, 10, 50, ….

c) 40, 48, 57.6, 69.12, ….

d) 4, 7, 16, 43, 124, ….

WB4 Find a recurrence relation for each of the following

a) 3, -7, 13, -27, -53, ….

b) 6, 12, 36, 164, 820, ….

c) 2, -4, 16, -256, ….

d) 1, 1

2 ,

1

3,

1

4,

1

5, ….


5

WB5 a sequence is defined by 𝑢𝑛+1 = 3𝑢𝑛 − 𝑐, and 𝑢1 = 4 where 𝑐 is a constant

a) Find an expression for 𝑢2 in terms of c

b) Given that 𝑢1 + 𝑢2 + 𝑢3 = 0 find the value of c

WB6 The nth term of a sequence is 𝑢𝑛, the sequence is defined by 𝑢𝑛+1 = 𝑝𝑢𝑛 + 𝑞, where 𝑝 & 𝑞 are constants The first three terms of the sequence are Find 𝑢1 = 2, 𝑢2 = 5 and 𝑢3 = 14

a) Show that 𝑞 = −1 and find the value of 𝑝 b) Find the value of 𝑢4


6

WB7 The nth term of a sequence is 𝑢𝑛 The sequence is defined by 𝑢𝑛+1 = 𝑝𝑢𝑛 + 𝑞 , where

p and q are constants

The first three terms are 𝑢1 = −10 𝑢2 = 11 𝑢3 = 0.5

a) Show that 𝑞 = 6 and find the value of p

b) find the value of 𝑢4

The limit of 𝑢𝑛 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 𝑢𝑛+1 = 2𝑢𝑛 + 3, and 𝑢1 = 𝑘 where 𝑘 is a positive integer a) Write an expression for 𝑢2 in terms of k b) Show that 𝑢3 = 4𝑘 + 9 c) Find 𝑢1 + 𝑢2 + 𝑢3 + 𝑢4 in terms of k, in its simplest form and show that the result is divisible by 3


7

SIGMA notation

WB9 Write the series

a) ∑ (3𝑟 + 5)𝑛𝑟=1

b) ∑ (3𝑟 + 5)10𝑟=1

c) ∑ (3𝑟 + 5)20𝑟=8

d) ∑ (3𝑟2 + 1)6𝑟=1


8

WB10 Write these series in sigma notation

a) 3 + 7 + 11 + 15 + 19 + ⋯ … + 79

b) 1 + 4 + 9 + 16 + 25 + ⋯ … + 10000

c) 76 + 69 + 62 + 55 + 48 + ⋯ … − 92

d) 2 + 16 + 54 + 128 + 250 + 432 + 686

WB11 Evaluate:

a) ∑ (𝑟2 + 1)8𝑟=5

b) ∑ (2𝑟)5𝑟=1

c) ∑ (−1)𝑟−18

𝑟=1 (3𝑟 − 1)

d) ∑ 10𝑟=4

𝑟

12−𝑟


9


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? 4½, 6, 7½, 9, 10½, …

11, 22, 33, 44, 55, …

0.4, 0.6, 0.8, 1.0, 1.2, …

14, 36, 58, 80, 102, …

-4, 10, 24, 38, 52, …

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 “go and add up the numbers from 1 to 100.” 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).


11

WB13 Structure of an arithmetic sequence: Structure of an arithmetic series:

𝑎, 𝑎 + 𝑑, 𝑎 + 2𝑑, 𝑎 + 3𝑑, … … . 𝑎 + (𝑛 − 2)𝑑, 𝑎 + (𝑛 − 1)𝑑

An arithmetic sequence has first term, a, common difference, d, number of terms, n Find the nth term of a sequence using 𝑎 + (𝑛 − 1)𝑑 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 …

The sum of n terms of a sequence is given by a Series:

𝑆𝑛 = ∑(𝑎 + (𝑟 − 1)𝑑) =

𝑛

𝑟=1

𝒂 + (𝒂 + 𝒅) + (𝒂 + 𝟐𝒅) + ⋯ … + [𝒂 + (𝒏 − 𝟏)𝒅]

𝒃) ∑(𝟒𝒓 + 𝟏𝟏) =

𝟓

𝒓=𝟏


12

WB14 Find the number of terms in each of these arithmetic series

𝟏𝟑 + 𝟏𝟕 + 𝟐𝟏 + 𝟐𝟓 + … … + 𝟗𝟑

𝟒𝟐 + 𝟑𝟗 + 𝟑𝟔 + 𝟑𝟑 + … … + (−𝟗𝟑)

𝟏𝟐

𝟕+ 𝟐 +

𝟏𝟗

𝟕+

𝟐𝟒

𝟕+ … … + 𝟏𝟕

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?


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 𝐿 = 𝑎 + (𝑛 − 1)𝑑

Show that your formula works!


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 2½ . Find the least value of n

for which the nth term exceeds 1000


15

You should be able to derive (give a proof) of the formula 𝑆𝑛 =𝑛

2[2𝑎 + (𝑛 − 1)𝑑]

in about six or seven clear steps with correct notation: LHS and RHS; and

short annotations to explain the steps


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


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?


18

WB23 Simultaneous equations

The first three terms of an arithmetic sequence are (4x – 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 10𝑎 + 45𝑑 = 113

b) Given the sixth term is 12, write a second equation in a and d

c) Find the values of a and d


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


20

Using Sigma notation

WB27 Evaluate

a) ∑ (7𝑟 − 3)461

b) ∑ (3𝑟 + 5)221

c) ∑ (4 + 10𝑟)211

WB28 Find the value of n such that :

a) ∑ (3𝑟 − 7) = 217𝑛𝑟=1

b) ∑ (65 − 4𝑟) = −882𝑛𝑟=1


21

WB29 Evaluate these: (𝑤ℎ𝑒𝑟𝑒 𝑟 ≠ 1)

𝑎) ∑(3𝑟)

30

𝑟=5

𝑏) ∑ (3𝑟 − 5)

40

𝑟=10

𝑐) ∑ (2𝑟 + 6)

38

𝑟=22

WB30 𝒔𝒉𝒐𝒘 𝒕𝒉𝒂𝒕 ∑(3𝑟 + 4)

𝑛

1

= 3 ∑ 𝑟

𝑛

1

+ 4𝑛


22

WB31

𝑒𝑣𝑎𝑙𝑢𝑎𝑡𝑒 ∑(2𝑟)

25

1

Evaluate these using the previous result

∑(2𝑟 + 3)

25

1

∑(4𝑟)

25

1

∑(4𝑟 − 7)

25

1

WB32 the nth term of an arithmetic sequence is (6𝑟 − 8)

a) Write down the first three terms of this sequence

b) State the value of the common difference

c) Show that ∑ (6𝑟 − 8) = 𝑛(3𝑛 − 5)𝑛𝑟=1


23

WB33 A sequence of terms 𝑢1, 𝑢2, 𝑢3 … . . Is defined by 𝑢𝑛 = 20 −4

5 𝑛

a) Write down the exact values of the value of 𝑢1, 𝑢2, and 𝑢3 b) find the value k such that of 𝑢𝑘 = 0

c) Find ∑ 𝑢𝑛19𝑛=1

WB34 a) In the arithmetic series 𝑘 + 2𝑘 + 3𝑘 + ⋯ + 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

𝑘

c) Find, in terms of k, the 60th term of the arithmetic sequence

(2k + 2), (4𝑘 + 4), (6𝑘 + 6), … giving your answer in simplest form

sequences and series part 1 · sequences and series part 1 5 wb5 a sequence is defined by 𝑛+1=3...

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