1. Geometric sequences A sequence in which you get from one term to the next by multiplying by a constant is called a geometric sequence. This is also known as a geometric progression (GP) and the constant multiplier is called the common ratio.  The first term of a GP is denoted by a.  Its common ratio is denoted by r.  Formula for the n th term of GP is ar n-1  n th term: u n = ar n-1

2. Examples Decide which of the following sequences are geometric progressions. If the sequence is GP then write the common ratio and the next term. (a) 3, 6, 12, 24, 48, 96 (d) 4, -12, 36, -108, 324, -972 (c) 3, 3.3, 3.63, 3.993, 4.3923 (b) 2, 2.4, 2.8, 3.2, 3.6, 4.00 (a) yes: Next term = 192r = 2, (b) no (c) yes: Next term = 4.83153r = 1.1, (d) yes: Next term = 2916r = -3,

3. Examples Write down the term indicated in square bracket in each of the following geometric sequences. (a) 1, 2, 4, 8, 16, [10 th term] (d) p, p 3, p 5, p 7, p 9, [ 9 th term] (c) 16, 8, 4, 2, 1, [ 8 th term] (b) 5, -10, 20, -40, 80, [ 8 th term] (a) a = 1, r =2, 10 th term =1 x 2 9 =512 (b) a =5, r =-2, 8 th term =5 x (-2) 7 = -640 (c) a =16, r = ½, 8 th term =16 x ( ½ ) 7 = 1 / 8 (d) a =p, r = p 2, 9 th term =p x ( p 2 ) 8 = p 17

4. Examples Find an expression for the n th term of each of the following GPs. (a) 1, 2, 4, 8, 16, 32 (d) p, p 3, p 5, p 7, p 9, (c) 16, 8, 4, 2, 1, (b) 5, -10, 20, -40, 80, (a) a = 1, r =2, n th term =2 n-1 (b) a =5, r =-2, n th term =5 x (-2) n-1 (c) a =16, r = ½, n th term =16 x ( ½ ) n-1 (d) a =p, r = p 2, n th term =p x ( p 2 ) n-1

5. Examples Find the number of terms in each of the following GPs. (a) 2, 10, 50, ….., 1250 (a) a = 2 x 5 n- 1 = 1250 2, r =5 5 n- 1 = 625 5 4 = 625 n – 1 = 4 n = 5 (b) 5, 20, 80, ……., 5120 n th term = (b) a = 5 x 4 n- 1 = 5120 5, r = 4 n- 1 = 1024 4 5 = 1024 n – 1 = 5 n = 6 n th term = 4 trial and improve ment

6. Examples Find the common ratio and the first term in these GPs. (a) the 2 nd tem is 15 and the 5 th term is 1875 (b) the 3 rd term is 6 and the 7 th term is 96 2 nd term = ar =(a) 5 th term =ar 4 =1875 [1] [2] [2]  [1] =r 3 =125giving r =5 Form [1] a x 5 = 15 giving a =3 3 rd term =ar 2 =(b) 7 th term =ar 6 =96 [1] [2] [2]  [1] =r 4 =16giving r =2 Form [1] a x 4 = 6 giving a =1.5 15 6