Transcript Slides: C1 Chapter 6 - Sequences

C1 Chapter 6 Arithmetic Series
Dr J Frost ([email protected])
Last modified: 7th October 2013
Types of sequences
common difference
𝑑
?
+3
+3
+3
2, 5, 8, 11, 14, …
This is a:
Arithmetic
? Series
common?ratio 𝑟
×2 ×2
×2
3, 6, 12, 24, 48, …
1, 1, 2, 3, 5, 8, …
? Series
Geometric
This is the Fibonacci Sequence. The
terms follow a recurrence relation
because each term?can be generated
using the previous ones.
The fundamentals of sequences
𝑈𝑛
𝑛
𝑛 = 3?
The 𝑛th? term.
If 𝑼𝒏 is the ‘current term’, how
could we describe:
?
The position.
The previous term:
The term before that:
𝑈3 = 8?
?
2, 5, 8, 11, 14, …
?
𝑼𝒏−𝟏
𝑼𝒏−𝟐
?
Thus the following sequence:
1, 1, 2, 3, 5, 8, …
Could be described using:
𝑼𝟏 = 𝟏,
𝑼𝟐 = 𝟏,
? + 𝑼𝒏−𝟐
𝑼𝒏 = 𝑼𝒏−𝟏
Term-to-term and position-to-term
2, 5, 8, 11, 14, 17, …
What is the formula for the 𝑛th term based on:
…the position of the term 𝒏:
𝑈𝑛 = 3𝑛 −?1
…the previous term:
𝑈𝑛 = 𝑈𝑛−1 + 3
?
𝑛≥1
𝑛th term of an arithmetic sequence
We often use 𝑎 to denote the first term. Recall that 𝑑 is the difference between
terms, and 𝑛 is the position of the term we’re interested in.
1st Term
𝑎 ?
2nd Term
𝑎 +?𝑑
3rd Term
𝑎 +?2𝑑
...
...
𝑈𝑛 = 𝑎 + 𝑛 − 1 𝑑
𝑛th term
𝑎 + (𝑛?− 1)𝑑
𝑛th term of an arithmetic sequence
Find the requested term of the following sequences.
2, 5, 8, 11, 14, 17, …
100th term
10, 8, 6, 4, …
50th term
𝑎 = 10,
? 𝑑 = −2,
? 𝑛 = 50
?
𝑈50 = −88
?
5𝑥, 𝑥, −3𝑥, −7𝑥, …
20th term
𝑎 = 5𝑥,
? 𝑑 = −4𝑥,
? 𝑛 = 20
?
𝑈20 = −71𝑥
?
𝑎 = 2,
? 𝑑 =?3, 𝑛 = 100
?
𝑈100 = 299
?
Give that the 3rd term of an arithmetic series is 20 and the 7th term is 12. Find
a) The first term.
b) The 20th term.
24?
−14
?
Exercises
1
The first term of an arithmetic sequence is 14. If the fourth term is 32, find the
common difference.
𝒅 =? 𝟔
2
Given that the 3rd term of an arithmetic series is 30 and the 10th term is 9, find 𝑎
and 𝑑.
𝒂 = 𝟑𝟔,?𝒅 = −𝟑
3
In an arithmetic series the 20th term is 14 and the 40th term is -6. Find the 10th
term.
𝟐𝟒
?
4
For which values of 𝑥 would the expression −8, 𝑥 2 and 17𝑥 form the first three
terms of an arithmetic series.
𝟏
𝒙= ?
,𝒙 = 𝟖
𝟐
The number of terms
Bro Tip: If you’re trying to work out the number of terms in a sequence, you can do
whatever you like to the terms in the sequence until you get 1 to 𝑛, after which the
number of terms becomes obvious.
1, 3, 5, 7, 9, … , 111
? … , 112
2, 4, 6, 8, 10,
1, 2, 3, 4,? 5, … , 56
So there are 56 terms.
Add or subtract such
that the numbers are
now multiples of the
common difference.
Then divide.
The number of terms
How many terms? (work out in your head!)
1
2
3
4
5
5, 10, 15, 20, … , 200
2, 5, 8, 11, 14, … , 449
9, 19, 29, 39, … , 1999
11, 16, 21, 26, … , 151
5, 9, 13, 17, … , 409
𝑛 = 40?
?
𝑛 = 150
?
𝑛 = 200
𝑛 = 29?
𝑛 = 102
?
Sum of the first 𝑛 terms of a sequence.
𝒏th term
𝑈𝑛 = 𝑎 + 𝑛 − 1 𝑑
sum of first 𝒏 terms
𝑛
𝑆𝑛 = 2𝑎 ?+ 𝑛 − 1 𝑑
2
Let’s prove it!
Find the sum of the first 30 terms of the following arithmetic sequences…
1
2 + 5 + 8 + 11 + 13 …
𝑆30 = 1365?
2
100 + 98 + 96 + ⋯
𝑆30 = 2130?
3
𝑝 + 2𝑝 + 3𝑝 + ⋯
𝑆30 = 465𝑝?
Bro Tips: Explicitly write out "𝑎 =
⋯ , 𝑑 = ⋯ , 𝑛 = ⋯”. You’re less
likely to plug in numbers wrong
into the formula.
Make sure you write 𝑆𝑛 = ⋯ so
you make clear to yourself (and
the examiner) that you’re finding
the sum of the first 𝑛 terms, not
the 𝑛th term.
Sum of the first 𝑛 terms of a sequence.
Find the greatest number of terms for the sum of 4 + 9 + 14 + ⋯ to exceed 2000.
𝑺𝒏 > 𝟐𝟎𝟎𝟎,
𝒂 = 𝟒,
𝒅=𝟓
𝒏
𝟐𝒂 + 𝒏 − 𝟏 𝒅 > 𝟐𝟎𝟎𝟎
𝟐
𝒏
𝟖 + 𝒏 − 𝟏 𝟓 > 𝟐𝟎𝟎𝟎
𝟐
?
𝟓𝒏𝟐 + 𝟑𝒏 − 𝟒𝟎𝟎𝟎 > 𝟎
𝒏 < −𝟐𝟖. 𝟓 𝒐𝒓 𝒏 > 𝟐𝟕. 𝟗
So 28 terms needed.
Exam Question
Edexcel C1 Jan 2012
𝑇 =?400
𝑃 = £24450
?
Exam Question
Exercise 6F
Q1a, c, e, g
Q2a, c
Q5, Q6, 8, 10
Using Σ
What do these summations mean?
10
2𝑛 = 2 + 4 + 6 + 8 +?⋯ + 18 + 20
𝑛=1
This is commonly seen
in exams.
4
𝑎𝑘 = 𝑎1 + 𝑎2 + 𝑎3 + ?
𝑎4
𝑘=1
15
10 − 2𝑘 = 0 + −2 + −4? + ⋯ + −20
𝑘=5
Using Σ
Bro Tip: As
always, start by
explicitly writing
out your 𝑎, 𝑛
and 𝑑 values.
20
4𝑟 + 1 = 860 ?
𝑟=1
5
10
3 + 2𝑞 = 48 ?
𝑞=0
3 − 𝑟 = −25 ?
𝑟=1
More on recurrence relations
There will occasionally be two series questions, one on nth term/sum of n terms, and the
other on recurrence relations. Note that the sequence may not be arithmetic.
Edexcel C1 May 2013 (Retracted)
How would you say
this in words?
𝑥2 =?1 − 𝑘
𝑥3 = 𝑥22 − 𝑘𝑥2
= 1−𝑘 2−𝑘 ?
1−𝑘
= 1 − 3𝑘 + 2𝑘 2
? 32
𝑘=
1
1
+1+ −
+⋯
2
2
1
= 50 × 1 + − × 50 = 25
2
= 1+ −
?
More on recurrence relations
Edexcel C1 Jan 2012
𝑥2 =?𝑎 + 5
𝑥3 = 𝑎 𝑎 +?5 + 5 = ⋯
𝑎2 + 5𝑎 + 5 = 41
𝑎2 + 5𝑎 − 36 = 0
𝑎 + 9 𝑎?− 4 = 0
𝑎 = −9 𝑜𝑟 4