Slides: C1 Chapter 6 - Sequences
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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 π ? +3 +3 +3 2, 5, 8, 11, 14, … This is a: Arithmetic ? Series common?ratio π ×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 ππ π π = 3? The πth? term. If πΌπ is the ‘current term’, how could we describe: ? The position. The previous term: The term before that: π3 = 8? ? 2, 5, 8, 11, 14, … ? πΌπ−π πΌπ−π ? Thus the following sequence: 1, 1, 2, 3, 5, 8, … Could be described using: πΌπ = π, πΌπ = π, ? + πΌπ−π πΌπ = πΌπ−π Term-to-term and position-to-term 2, 5, 8, 11, 14, 17, … What is the formula for the πth term based on: …the position of the term π: ππ = 3π −?1 …the previous term: ππ = ππ−1 + 3 ? π≥1 πth term of an arithmetic sequence We often use π to denote the first term. Recall that π is the difference between terms, and π is the position of the term we’re interested in. 1st Term π ? 2nd Term π +?π 3rd Term π +?2π ... ... ππ = π + π − 1 π πth term π + (π?− 1)π πth 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 π = 10, ? π = −2, ? π = 50 ? π50 = −88 ? 5π₯, π₯, −3π₯, −7π₯, … 20th term π = 5π₯, ? π = −4π₯, ? π = 20 ? π20 = −71π₯ ? π = 2, ? π =?3, π = 100 ? π100 = 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. π =? π 2 Given that the 3rd term of an arithmetic series is 30 and the 10th term is 9, find π and π. π = ππ,?π = −π 3 In an arithmetic series the 20th term is 14 and the 40th term is -6. Find the 10th term. ππ ? 4 For which values of π₯ would the expression −8, π₯ 2 and 17π₯ form the first three terms of an arithmetic series. π π= ? ,π = π π 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 π, 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 π = 40? ? π = 150 ? π = 200 π = 29? π = 102 ? Sum of the first π terms of a sequence. πth term ππ = π + π − 1 π sum of first π terms π ππ = 2π ?+ π − 1 π 2 Let’s prove it! Find the sum of the first 30 terms of the following arithmetic sequences… 1 2 + 5 + 8 + 11 + 13 … π30 = 1365? 2 100 + 98 + 96 + β― π30 = 2130? 3 π + 2π + 3π + β― π30 = 465π? Bro Tips: Explicitly write out "π = β― , π = β― , π = β―”. You’re less likely to plug in numbers wrong into the formula. Make sure you write ππ = β― so you make clear to yourself (and the examiner) that you’re finding the sum of the first π terms, not the πth term. Sum of the first π terms of a sequence. Find the greatest number of terms for the sum of 4 + 9 + 14 + β― to exceed 2000. πΊπ > ππππ, π = π, π =π π ππ + π − π π > ππππ π π π + π − π π > ππππ π ? πππ + ππ − ππππ > π π < −ππ. π ππ π > ππ. π So 28 terms needed. Exam Question Edexcel C1 Jan 2012 π =?400 π = £24450 ? Exam Question Exercise 6F Q1a, c, e, g Q2a, c Q5, Q6, 8, 10 Using Σ What do these summations mean? 10 2π = 2 + 4 + 6 + 8 +?β― + 18 + 20 π=1 This is commonly seen in exams. 4 ππ = π1 + π2 + π3 + ? π4 π=1 15 10 − 2π = 0 + −2 + −4? + β― + −20 π=5 Using Σ Bro Tip: As always, start by explicitly writing out your π, π and π values. 20 4π + 1 = 860 ? π=1 5 10 3 + 2π = 48 ? π=0 3 − π = −25 ? π=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? π₯2 =?1 − π π₯3 = π₯22 − ππ₯2 = 1−π 2−π ? 1−π = 1 − 3π + 2π 2 ? 32 π= 1 1 +1+ − +β― 2 2 1 = 50 × 1 + − × 50 = 25 2 = 1+ − ? More on recurrence relations Edexcel C1 Jan 2012 π₯2 =?π + 5 π₯3 = π π +?5 + 5 = β― π2 + 5π + 5 = 41 π2 + 5π − 36 = 0 π + 9 π?− 4 = 0 π = −9 ππ 4
