Arithmetic Sequences Geometric Sequences What types of sequences are there? 2,6,10,14,……… 3,6,12,24,48,… 16,9,2,-5,……… 63,21,7, ,…… Both of these sequences are made by…… Both of these sequences are made by…… Multiplying! Adding! (4 in the first one) (-7 in the 2nd one) (by 2 in the first one) (by in the 2nd one) These ones are called………. These ones are called………. Arithmetic Sequences Geometric Sequences

Arithmetric Sequences Geometric Sequences 2,6,10,14,……… 3,6,12,24,48,… u1 u2 u3 u4 u1 u2 u3 u4 This sequence can be re-written as….. This sequence can be re-written as….. 2,2+14,2+24,2+34,… 3,321,322,323,… u2 u1 u4 u3 u2 u1 u4 u3 Now the sequences are written in terms of the first term and the amount by which the terms are changing This will allow us to calculate other terms in the sequence, without writing them all out

Arithmetric Sequences Geometric Sequences 2,6,10,14,……… 3,6,12,24,48,… Speaking in general terms……... If the first term is a and the constant difference is d, then the sequence is…….. If the first term is a and the constant quantity is r, then the sequence is…….. {a,a + d,a + 2d,a + 3d…….. } {a,ar,ar2,ar3,ar4…….. } The amount by which the sequence changes (called the common difference) is found by….. The amount by which the sequence changes(called the common ratio) is found by….. d = u2 - u1 = u3 - u2 = u4 - u3 r = u2  u1 = u3  u2 = u4  u3 And, if u1,u2 & u3 are 3 terms in a sequence, then….. And, if u1,u2 & u3 are 3 terms in a sequence, then….. u2 - u1 = u3 - u2 u2  u1 = u3  u2  i.e. 2u2 = u1 + u3 u2 = ½(u1 + u3) i.e. u22 = u1  u3

Arithmetric Sequences Geometric Sequences {a,a + d,a + 2d,a + 3d…….. } {a, ar, ar2, ar3, ar4…….. } u1 u2 u3 u4 u1 u2 u3 u4 What do you notice about the number of each term compared to the number of d’’s and r’s? The number of d’’s & r’s is always one less than the term! So the general rule for find any term is un = a + (n - 1)d un = ar(n-1)

All these HAVE to be remembered! To summarise…….. Arithmetic Sequence (A.P.) Geometric Sequence (G.P.) 1. Sequence is {a,a+d,a+2d,a+3d…} 1. Sequence is {a,ar,ar2, ar3, …} 2. General rule for A.P. un= a + (n - 1)d 2. General rule for G.P. un= ar(n-1) 3. Test for A.P. d = u2 - u1 = u3 - u2 = …….. 3. Test for G.P. r = u2  u1 = u3  u2 = …….. 4. If a, b & c are in an A.P. then b - a = c - b, i.e. 2b = a + c 4. If a, b & c are in an A.P. then b  a = c  b, i.e. b2 = ac All these HAVE to be remembered! 

Some examples: 1. Find the tenth term of the following sequences: G.P. b) log4, log8, log16, log32, …….. This sequence must be a ……... This is a VERY POPULAR question G.P. Log4 = log22 = 2log2 because r = 8  4 = 16  8 = 2 log8 = log23 = 3log2 With a = 4 log16 = log24 = 4log2 So,as un= ar(n-1) log32= log25 = 5log2 u10= 4  2(10-1) So the sequence is really…... 2log2, 3log2, 4log2, 5log2, ………. u10= 22  29 This sequence must be ……... u10= 211 an A.P. because d = 3log2 - 2log2 u10= 2048 because d = log2 So,as un= a + (n - 1)d un= 2log2 + (10 - 1) log2 un= 11log2 = log211 = log2048