C1 Chapter 6

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This content delves into arithmetic sequences, including types of sequences, fundamentals, term-to-term relationships, and finding specific terms in a sequence. It covers concepts like common differences, positions of terms, and solving for unknown terms in arithmetic series.

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C1 Chapter 6 Arithmetic Series Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 7th October 2013

Types of sequences common difference ? ? +3 +3 This is a: +3 2, 5, 8, 11, 14, Arithmetic Series ? ? common ratio ? 2 2 2 3, 6, 12, 24, 48, Geometric Series ? This is the Fibonacci Sequence. The terms follow a recurrence relation because each term can be generated using the previous ones. ? 1, 1, 2, 3, 5, 8,

The fundamentals of sequences If ??is the current term , how could we describe: ?? The ?thterm. ? ? ?? ? ?? ? ? The previous term: The term before that: ? The position. ? ? ? = 3 ? ?3= 8 ? Thus the following sequence: 1,1,2,3,5,8, Could be described using: 2,5,8,11,14, ??= ?, ??= ?? ?+ ?? ? ??= ?, ?

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 ?: ? ??= 3? 1 the previous term: ? ??= ?? 1+ 3 ? 1

?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 ? ? ? + 2? ? ? + (? 1)? ? ? ? + ? . . . ??= ? + ? 1 ?

?th term of an arithmetic sequence Find the requested term of the following sequences. ? ? ? 100th term 2,5,8,11,14,17, ? = 2,? = 3,? = 100 ?100= 299 ? ? ? ? ? = 10,? = 2,? = 50 ?50= 88 ? ? ? 50th term 10,8,6,4, 5?,?, 3?, 7?, 20th term ? ? ? = 5?,? = 4?,? = 20 ?20= 71? Give that the 3rd term of an arithmetic series is 20 and the 7th term is 12. Find ? 24 14 ? a) The first term. b) The 20th term.

Exercises The first term of an arithmetic sequence is 14. If the fourth term is 32, find the common difference. ? = ? 1 ? Given that the 3rd term of an arithmetic series is 30 and the 10th term is 9, find ? and ?. ? = ??,? = ? 2 ? In an arithmetic series the 20th term is 14 and the 40th term is -6. Find the 10th term. 3 ?? ? For which values of ? would the expression 8,?2 and 17? form the first three terms of an arithmetic series. ? =? ?,? = ? 4 ?

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 ?, after which the number of terms becomes obvious. 1,3,5,7,9, ,111 Add or subtract such that the numbers are now multiples of the common difference. 2,4,6,8,10, ,112 ? 1,2,3,4,5, ,56 So there are 56 terms. Then divide. ?

The number of terms How many terms? (work out in your head!) ? ? ? ? ? ? = 40 ? = 150 ? = 200 ? = 29 ? = 102 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 1 2 3 4 5

Sum of the first ? terms of a sequence. ?th term sum of first ? terms ??=? 2? + ? 1 ? ? ??= ? + ? 1 ? 2 Let s prove it! Find the sum of the first 30 terms of the following arithmetic sequences ? 2 + 5 + 8 + 11 + 13 ?30= 1365 1 Bro Tips: Explicitly write out "? = ,? = ,? = . You re less likely to plug in numbers wrong into the formula. ? 100 + 98 + 96 + ?30= 2130 2 ? ? + 2? + 3? + ?30= 465? 3 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 4 + 9 + 14 + to exceed 2000. ??> ????, ? ??? + ? ? ? > ???? ? ?? + ? ? ? > ???? ???+ ?? ???? > ? ? < ??.? ?? ? > ??.? So 28 terms needed. ? = ?, ? = ? ?

Exam Question Edexcel C1 Jan 2012 ? = 400 ? ? = 24450 ?

Exam Question Exercise 6F Q1a, c, e, g Q2a, c Q5, Q6, 8, 10

Using What do these summations mean? 10 ? 2? = 2 + 4 + 6 + 8 + + 18 + 20 ?=1 This is commonly seen in exams. 4 ? ??= ?1+ ?2+ ?3+ ?4 ?=1 15 ? 10 2? = 0 + 2 + 4 + + 20 ?=5

Using Bro Tip: As always, start by explicitly writing out your ?,? and ? values. 20 4? + 1 = 860 ? ?=1 5 10 ? ? 3 + 2? = 48 3 ? = 25 ?=0 ?=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? ?2= 1 ? ? 2 ??2 ?3= ?2 = 1 ?2 ? 1 ? = 1 3? + 2?2 ? ? =3 ? 2 = 1 + 1 + 1 + 1 ? + 2 2 = 50 1 + 1 2 50 = 25