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Sequences- Nth Term Skipton Girlsβ High School Objectives: Understand term-to-term vs position-to-term rules. Be able to generate terms of a sequence given a formula. Find the formula for a linear sequence. Be able to find a term of an oscillating sequence. Last modified: 9th June 2016 STARTER :: Whatβs next in each sequence? A sequence is simply an ordered list of items (possibly infinitely long), usually with some kind of pattern. What are the next two terms in each sequence? a b c d e f g h ? β¦ 6, 13, 20, 27, ππ, ππ, 1 π ? β¦ 4, 2 , 1, β , βπ, 2 π ? 4, 12, 36, πππ, πππ, β¦ 4, 6, 9, 13, ππ, ππ,? β¦ ? β¦ 2, 5, 7, 12, 19, ππ, ππ, ? 5, 25, 15, 75, 65, πππ, πππ, β¦ ? 1, 8, 27, 64, πππ, πππ, β¦ 243 ÷ 27 = 9, 27 ÷ 9 = 3 243, 27, 9, 3, 3, π, π β¦ ? And so on. (Nicked off 2015βs βChild Geniusβ on Channel 4) Term-to-term rules Some sequences we can generated by stating a rule to say how to generate the next term given the previous term(s). Description First 5 terms The first term of a sequence is 1. +3 to each term to get the next. 1, 4, 7, 10, 13 ? The first term of a sequence is 3. × 2 to each term to get the next. 3, 6, 12, 24, 48 ? The first two terms are 0 and 1. Add the last two terms to get the next. 0, 1, 1, 2, 3 ? (known as the Fibonacci sequence) What might be the disadvantage of using a term-to-term rule? To get a particular term in the sequence, we have to generate all the terms in the sequence before it. This ? is rather slow if you say want to know the 1000th term! JMC Puzzle [JMC 2009 Q11] In a sequence of numbers, each term after the first three terms is the sum of the previous three terms. The first three terms are -3, 0, 2. Which is the first term to exceed 100? A 11th term B 12th term C 13th term D 14th term E 15th term A B C D Terms are: -3, 0, 2, -1, 1, 2, 2, 5, 9, 16, 30, 55, 101 E Position-to-term :: βπth termβ Itβs sometimes more helpful to be able to generate a term of a formula based on its position in the sequence. We could use it to say find the 300th term of a sequence without having to write all the terms out! We use π to mean the position in the sequence. So if we want the 3rd term, π = 3. πth term ππ§ ππ§ ππ§ β π π§π + π π§ π§+π π ππ 1st term ?3 ?5 ?1 ?2 ?1 ?2 2nd term 6 ? 10 ? 3 ? 5 ? 3 ? 4 ? 3rd term ?9 15 ? ?5 10 ? ?6 ?8 4th term 12 ? 20 ? ?7 17 ? 10 ? 16 ? So 3π gives the 3 times table, 5π the five times table, and so on. This formula gives the triangular numbers! Check Your Understanding Find the first 4 terms in each of these sequences, given the formula for the πth term. 4π + 3 3π β 2 π2 β π 2π + 3π β β β β π, ππ, ππ,?ππ π, π, π, ππ? π, π, π, ππ? π, ππ, ππ,?ππ Exercise 1 1 Find the 100th [First part of JMO 2001 B2] In a sequence, each 5 term of the sequences with the term after the first is the sum of the squares of the following formulae for the πth term: a) 8π β 3 797 b) 3 β π -97 2 c) 3π β π + 1 29901 digits of the previous term. Thus if the first term were 12, the second term would be 12 + 22 = 5, the third term 52 = 25, the fourth term 22 + 52 = 29, and so on. Find the first five terms of the sequence whose first term is 25. 25, 29, 85, 89, 145 ? ? ? 2 A sequence starts with 1. Thereafter, each new ? term is formed by adding all the previous terms, and then adding 1. What are the first 6 terms? [First part of JMO 2005 B1] The first three terms of 6 1 1 1 1 1 1 1, 2, 4, 8, 16, 32 a sequence are , , . The fourth term is β + ; 4 3 2 2 3 4 henceforth, each new term is calculated by taking 3 Find the first 4 terms of the following the previous term, subtracting the term before that, sequences: and then adding the term before that. a) π + 3 4, 5, 6, 7 Write down the first six terms of the sequence, b) 3π 3, 9, 27, 81 giving your answers as simplified fractions. c) π3 β π2 0, 4, 18, 48 ? d) e) ? ? ? -2, -3, -2, ? 1 π2 β 4π + 1 π! (Look for it on your calculator) 1, 2, 6, 24 4 [JMC 2014 Q11] The first two terms of a ? π π π π π π , , , , , π π π ππ π π ? N [JMO 2010 B1] In a sequence of six numbers, every sequence are 1 and 2. Each of the following terms in the sequence is the sum of all the terms which come before it in the sequence. Which of these is not a term in the sequence? A 6 B 24 C 48 D 72 E 96 Solution: D term after the second term is the sum of the previous two terms. Also, the last term is four times the first term, and the sum of all six terms is 13. (Hint: perhaps represent What is the first term? π the first two terms Solution: π ? π algebraically?) Linear Sequences Todayβs title What sequence does 5π give? π, ππ, ππ, ? ππ, β¦ What therefore would 5π β 4 give? π, π, ππ,? ππ, β¦ What do you notice about the difference between terms in this sequence? It goes up by 5 ?each time. What therefore do you think would be the difference between terms for: 6π + 2 πβ1 10π β 3 3βπ β6 β1 β 10 β β1 ? ? ? ? Finding πth term formula for linear sequences Find the πth term of the following sequence: 5, 9, 13, 17, 21 β¦ 4π + 1 ? ? We saw that the number on front of the π gives us the (first) difference between terms. If we had 4π as our formula, this would give us the 4 times table. So what βcorrectionβ is needed? Note: Why do you think this is known as a βlinearβ sequence? If you plotted each position with the term on some axes (e.g. for this sequence (1,5),(2,9),(3,13),(4,17), β¦, ? it would form a straight line. The word βlinearβ means βstraightβ. More examples 7, 12, 17, 22, 27, β¦ 5, 7, 9, 11, 13, β¦ 2, 5, 8, 11, 14, β¦ 4, 10, 16, 22, 28, β¦ 10, 8, 6, 4, 2, β¦ β β β β β ? ππ + π ? ππ + π ππ β π ? ππ β π ? βππ + ππ (or ? ππ β ππ) Quickfire Questions: πth term: 5, 8, 11, 14, 17, β¦ 3, 9, 15, 21, 27, β¦ 9, 14, 19, 24, 29, β¦ 2, 9, 16, 23, 30, β¦ β β β β 3π +? 2 6π β? 3 5π +? 4 7π β? 5 100th term: ? 302 ? 597 504 ? 695 ? Test Your Understanding πth term: 10, 18, 26, 34, β¦ 2, 8, 14, 20, 26, β¦ 10, 9, 8, 7, 6, β¦ 1 1 3 , 5, 6 , 8, β¦ 2 2 β 8π +? 2 β 6π β? 4 β 11 β?π 3 β π +?2 2 100th term: ? 802 ? 596 ? β89 152 ? Is a number in the sequence? Is the number 598 in the sequence with πth term 3π β 2? Could we obtain 598 using the ππ β π formula? ? π = πππ. So 598 is the Yes! Working backwards, we see 200th term in the sequence. Is the number 268 in the sequence with πth term 4π β 2? No. ππ β π = πππ ? But adding 2 we get 270, and 270 is not divisible by 4. Exercise 2 1 Find the πth term and the 300th term of the following sequences. a b c d e f 5, 8, 11, 14, β¦ 4, 11, 18, 25, β¦ 11, 16, 21, 26, β¦ 6, 17,28,39, β¦ 16,20,24,28, β¦ 9,32,55,78, β¦ 1 1 1, 1 2 , 2, 2 2 , β¦ g 2 β ππ + π,?πππ β ππ β π,?ππππ β ππ + π,?ππππ β πππ β π, ? ππππ β ππ + ππ, ? ππππ β πππ β ππ, ππππ ? π π π β ππ + π? , πππ π Determine (with working) whether the following numbers are in the sequence with the πth term formula. If so, indicate the position of the term. Yes (6th ? term) No ? st Yes (31 ?term) No ? a b c d 30 in 5π 90 in 3π + 2 184 in 6π β 2 148 in π2 + 2 3 Find the missing numbers in these linear sequences. a b 3, ? , ? , ? , 19 π, ππ, ? ππ 4, ? , ? , ? , ? , 10 (π. π, π. π,?π. π, π. π) 4 Find the formula for the πth term of the following sequences. a 6, 5, 4, 3, 2, β¦ πβπ π β ππ b 5, 2, β1, β4, β¦ 1 1 π ππ β π c 10 , 8, 5 , 3, β¦ 2 2 π 1 7 5 1 π ππ d 2 ,2 ,2 ,3 π+ 3 12 6 12 π ππ ? ? ? ? 5 The 3rd term of a linear sequence is 17. The 45th term is 269. Determine the formula for the πth term. ππ β π ? Two sequences have the formulae 3π β 1 and N 7π + 2. A new sequence is formed by the numbers which appear in both of these sequences. Determine the formula for the πth term. πππ + π Whatever the first number is that coincides, weβll see it 21 later because this is the βlowest common multipleβ of 3 and 7. Thus we know the formula is of the form πππ + β‘. Itβs then simply a case of identifying which number this is (2). This principle is known as the βChinese Remainder Theoremβ. ?
