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GCSE: L inear Sequences

GCSE: L inear Sequences. Dr J Frost (jfrost@tiffin.kingston.sch.uk) . Last modified: 19 th January 2013. Recap: Formula for n th term. Either find the formula for the n th term, or given the formula, find the first few terms of the formula. ?. ?. ?. ?. ?. ?. ?. ?. ?. ?.

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GCSE: L inear Sequences

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  1. GCSE: Linear Sequences Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 19th January 2013

  2. Recap: Formula for nth term Either find the formula for the nth term, or given the formula, find the first few terms of the formula. ? ? ? ? ? ? ? ? ? ?

  3. Exercise 1 Fill in the missing parts of the table. ? ? 1 2 ? ? ? 3 ? 4 ? ? 5 ? ? 6 ? ? 7 ? ? 8 ? ? 9 ? ?

  4. Interpreting Sequences There are often follow up questions which require more than just computing a given term of the sequence or the formula for the nth term. Q 3, 7, 11, 15, 19, ... Jasmine argues that 204 is in the sequence. Is she right? No, because 204 is an even number, and the sequence contains only odd numbers. ? Q 4, 7, 10, 13, 16, ... Tarquin argues that 3002 is in the sequence. Is he right? No, because 3002 is two more than a multiple of 3, whereas the numbers in the sequence are 1 more than a multiple of 3. ?

  5. Difficult Sequences Question (From a GCSE specimen paper) Q 1, 4, 7, 10, ... Find an expression for the nth term of the sequence.3n – 2 Sophie claims that when she squares any term in the sequence, she gets a term that is in the same sequence. Show that she is right.(3n – 2)2 = 9n2 – 12n + 4 = 9n2 – 12n + 6 – 2 = 3(3n2 – 4n + 2) – 2 This is in the form “3 times an integer minus 2”, so must be in the sequence. ? The approach amounts to squaring the formula for the nth term, and showing (by rearranging) that this can written in the same form (i.e. three times something minus 2).

  6. Difficult Sequences Question Now your turn... Q 7, 13, 19, 25, 31, ... Find an expression for the nth term of the sequence.6n + 1 Bob claims that when he squares any term in the sequence, he gets a term that is in the same sequence. Show that he is right.(6n + 1)2 = 36n2 + 12n + 1 = 6(6n2 + 2n) + 1 This is one more than a multiple of 6, so must be in the sequence. ? ?

  7. Exercise 2 2 Prove algebraically that the following actions to any term in the sequence gives a term still in the sequence. For the following, justify why the given term is not in the sequence. 1 5, 9, 13, 17, ... Squaring (4n + 1)2 = 16n2 + 8n + 1 = 4(n2 + 2n) + 1 1, 6, 11, 16, ... Adding 15. (5n – 4) + 15 = 5n + 11 = 5n + 15 – 4 = 5(n + 3) – 4 1, 7, 13, 19, 25, ... Squaring (6n – 5)2 = 36n2 – 60n + 25 = 36n2 – 60n + 30 – 5 = 6(6n2 – 10n + 10) - 5 a ? a b ? b c ? d ? c e ?

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