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Types of sequences
2, 5, 8, 11, 14, …+3 +3 +3 This is a:
Arithmetic Series?
Geometric Series?3, 6, 12, 24, 48, …×2 ×2 ×2
common difference
common ratio
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1, 1, 2, 3, 5, 8, …This is the Fibonacci Sequence. The terms follow a recurrence relation because each term can be generated using the previous ones.
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The fundamentals of sequences
𝑈𝑛 The th term.
𝑛 The position.
2 ,5 ,8 ,11 ,14 ,…
𝑛=3 𝑈 3=8
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If is the ‘current term’, how could we describe:
The previous term:The term before that:
Thus the following sequence:
Could be described using:
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Term-to-term and position-to-term
2 ,5 ,8 ,11 ,14 ,17 ,…What is the formula for the th term based on:
…the position of the term :
…the previous term:
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th term of an arithmetic sequence
We often use to denote the first term. Recall that is the difference between terms, and is the position of the term we’re interested in.
1st Term 2nd Term 3rd Term th term. . .
𝑎 𝑎+𝑑 𝑎+2𝑑 𝑎+(𝑛−1)𝑑? ? ? ?
𝑈𝑛=𝑎+(𝑛−1 )𝑑
. . .
th 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
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.
5 𝑥 , 𝑥 ,−3 𝑥 ,−7 𝑥 ,… 20th term
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Exercises
The first term of an arithmetic sequence is 14. If the fourth term is 32, find the common difference.
Given that the 3rd term of an arithmetic series is 30 and the 10th term is 9, find and .
In an arithmetic series the 20th term is 14 and the 40th term is -6. Find the 10th term.
For which values of would the expression and form the first three terms of an arithmetic series.
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1
2
3
4
The number of termsBro 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 to , after which the number of terms becomes obvious.
1 ,3 ,5 ,7 ,9 ,…,111
2 ,4 ,6 ,8 ,10 ,…,112
So there are 56 terms.
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Add or subtract such that the numbers are now multiples of the common difference.
Then divide.
The number of termsHow many terms? (work out in your head!)
5, 10, 15, 20, … , 2002, 5, 8, 11, 14, … , 4499, 19, 29, 39, … , 199911, 16, 21, 26, … , 1515, 9, 13, 17, … , 409
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2
3
4
5
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Sum of the first terms of a sequence.
𝑈𝑛=𝑎+(𝑛−1 )𝑑 𝑆𝑛=𝑛2 (2𝑎+ (𝑛−1 )𝑑 )
th term sum of first terms
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Let’s prove it!
Find the sum of the first 30 terms of the following arithmetic sequences…
1
2
3
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Bro Tips: Explicitly write out ”. You’re less likely to plug in numbers wrong into the formula.
Make sure you write so you make clear to yourself (and the examiner) that you’re finding the sum of the first terms, not the th term.
Sum of the first terms of a sequence.
Find the greatest number of terms for the sum of to exceed 2000.
So 28 terms needed.
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Edexcel C1 Jan 2012
𝑇=400
𝑃=£ 24450
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Exam Question
Exam Question
Exercise 6FQ1a, c, e, gQ2a, cQ5, Q6, 8, 10
Using What do these summations mean?
∑𝑛=1
10
2𝑛=2+4+6+8+…+18+20
∑𝑘=1
4
𝑎𝑘=𝑎1+𝑎2+𝑎3+𝑎4
∑𝑘=5
15
(10−2𝑘)=0+ (−2 )+(−4 )+…+ (−20 )
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This is commonly seen in exams.
Using Bro Tip: As always, start by explicitly writing out your and values.
∑𝑟=1
20
4 𝑟+1=860
∑𝑟=1
10
3−𝑟=−25∑𝑞=0
5
3+2𝑞=48
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More on recurrence relationsThere 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
𝑘=32
𝑥3=𝑥22−𝑘𝑥2
¿1+(− 12 )+1+(− 12 )+…
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More on recurrence relationsEdexcel C1 Jan 2012
𝑥2=𝑎+5
𝑥3=𝑎 (𝑎+5 )+5=…
𝑎2+5𝑎+5=41
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![Arithmetic sequences and series[1]](https://static.documents.pub/doc/80x56/5489af66b479590f0d8b5953/arithmetic-sequences-and-series1.jpg)
