A Day in the Life of a Teacher by Heather Adamson

By Heather Adamson

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A Show that the sequence is arithmetic. b Find the formula for the general term un . c Find the 100th term of the sequence. d Is i 828 ii 2341 a member of the sequence? 9¡2 = 7 16 ¡ 9 = 7 23 ¡ 16 = 7 30 ¡ 23 = 7 a So, assuming that the pattern continues, consecutive terms differ by 7. ) the sequence is arithmetic with u1 = 2, d = 7. , un = 7n ¡ 5 ii Let un = 2341 ) 7n ¡ 5 = 2341 ) 7n = 2346 ) n = 335 17 Let un = 828 7n ¡ 5 = 828 ) 7n = 833 ) n = 119 ) 828 is a term of the sequence. In fact it is the 119th term.

CDR Wednesday, 24 October 2007 12:47:21 PM PETERDELL IB_HL-2ed 58 SEQUENCES AND SERIES (Chapter 2) Example 4 Find the general term un for an arithmetic sequence with u3 = 8 and u8 = ¡17: ) ) u3 = 8 u8 = ¡17 u1 + 2d = 8 u1 + 7d = ¡17 fun = u1 + (n ¡ 1)dg :::: (1) :::: (2) We now solve (1) and (2) simultaneously: ¡u1 ¡ 2d = ¡8 u1 + 7d = ¡17 ) 5d = ¡25 ) d = ¡5 fadding the equationsg u1 + 2(¡5) = 8 ) u1 ¡ 10 = 8 ) u1 = 18 So in (1) Now ) ) ) un un un un Check: u3 = 23 ¡ 5(3) = 23 ¡ 15 =8 X = u1 + (n ¡ 1)d = 18 ¡ 5(n ¡ 1) = 18 ¡ 5n + 5 = 23 ¡ 5n u8 = 23 ¡ 5(8) = 23 ¡ 40 = ¡17 X Example 5 Insert four numbers between 3 and 12 so that all six numbers are in arithmetic sequence.

B Find un and hence find the 15th term. 0 2 5 95 100 50 75 25 0 5 1 For the geometric sequence with first two terms given, find b and c: a 2, 6, b, c, .... b 10, 5, b, c, ..... c 12, ¡6, b, c, ..... CDR Monday, 22 October 2007 4:02:54 PM PETERDELL IB_HL-2ed 62 SEQUENCES AND SERIES (Chapter 2) 4 Show that the sequence 8, ¡6, 4:5, ¡3:375, .... is geometric and hence find the 10th term as a decimal. p p 5 Show that the sequence 8, 4 2, 4, 2 2, .... is geometric. Hence find, in simplest form, the general term un .

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