The possible substitutions fornare: n=1, 2 , 3 , 4 , 5 , 6 , ...... so an algebraic rule forunis valid for
n 2 Z+only.Example 3 Self Tutor
Find the first 5 terms of the sequence with the rule:
a un=5n¡ 3 b un=n(n+2) c un=3£ 2 na u 1 = 5(1)¡3=2
u 2 = 5(2)¡3=7
u 3 = 5(3)¡3=12
u 4 = 5(4)¡3=17
u 5 = 5(5)¡3=22
The sequence is:
2 , 7 , 12 , 17 , 22 , ......b u 1 = 1(3) = 3
u 2 = 2(4) = 8
u 3 = 3(5) = 15
u 4 = 4(6) = 24
u 5 = 5(7) = 35
The sequence is:
3 , 8 , 15 , 24 , 35 , ......c u 1 = 3(2)^1 =6
u 2 = 3(2)^2 =12
u 3 = 3(2)^3 =24
u 4 = 3(2)^4 =48
u 5 = 3(2)^5 =96
The sequence is:
6 , 12 , 24 , 48 , 96 , ......Discussion Properties of sequences
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² Consider the sequence inExample 3parta. The sequence islinearbecause each term differs from
the previous one by the same constant 5.
What part of the formula forunindicates this fact?
² What can be said about the formula for a linear sequence where each term differs from the previous
one by: a 7 b ¡ 4?
² Consider the sequence inExample 3partc. Notice that each term is double the previous term.
What part of the formula foruncauses this?Example 4 Self Tutor
a Find the next two terms and an expression for thenth termunof 3 , 6 , 9 , 12 , 15 , ......
b Hencefind a formula for the general termunof:
i 4 , 7 , 10 , 13 , 16 , ...... ii 1 , 4 , 7 , 10 , 13 , ...... iii^15 ,^18 , 111 , 141 , ......a u 1 =3£ 1 , u 2 =3£ 2 , u 3 =3£ 3 , u 4 =3£ 4 , u 5 =3£ 5
) un=3£n=3n
) u 6 =3£6=18and u 7 =3£7=21:
biu 1 =3+1, u 2 =6+1, u 3 =9+1, u 4 =12+1, u 5 =15+1
Each term is 1 more than in the sequence ina.
) un=3n+1
ii u 1 =3¡ 2 , u 2 =6¡ 2 , u 3 =9¡ 2 , u 4 =12¡ 2 , u 5 =15¡ 2
Each term is 2 less than in the sequence ina.
) un=3n¡ 2iii u 1 =1
3+2
, u 2 =1
6+2
, u 3 =1
9+2
, u 4 =1
12 + 2
, u 5 =1
15 + 2
By comparison with the sequence ina, un=1
3 n+2536 Sequences (Chapter 26)IGCSE01
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Y:\HAESE\IGCSE01\IG01_26\536IGCSE01_26.CDR Monday, 27 October 2008 2:36:06 PM PETER