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Geometric sequences

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.

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Geometric sequences

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  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 nth term of GP is arn-1  nth term: un = arn-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 (b) 2, 2.4, 2.8, 3.2, 3.6, 4.00 (c) 3, 3.3, 3.63, 3.993, 4.3923 (d) 4, -12, 36, -108, 324, -972 (a) yes: r = 2, Next term = 192 (b) no (c) yes: r = 1.1, Next term = 4.83153 (d) yes: r = -3, Next term = 2916

  3. Examples Write down the term indicated in square bracket in each of the following geometric sequences. (a) 1, 2, 4, 8, 16, [10th term] (b) 5, -10, 20, -40, 80, [ 8th term] (c) 16, 8, 4, 2, 1, [ 8th term] (d) p, p3, p5, p7, p9, [ 9th term] (a) a = 1, r = 2, 10th term = 1 x 29 = 512 (b) a = 5, r = -2, 8th term = 5 x (-2)7 = -640 (c) a = 16, r = ½ , 8th term = 16 x ( ½ )7 = 1/8 (d) a = p, r = p2 , 9th term = p x ( p2 )8 = p17

  4. Examples Find an expression for the nth term of each of the following GPs. (a) 1, 2, 4, 8, 16, 32 (b) 5, -10, 20, -40, 80, (c) 16, 8, 4, 2, 1, (d) p, p3, p5, p7, p9, (a) a = 1, r = 2, nth term = 2n-1 (b) a = 5, r = -2, nth term = 5 x (-2)n-1 (c) a = 16, r = ½ , nth term = 16 x ( ½ )n-1 (d) a = p, r = p2 , nth term = p x ( p2 )n-1

  5. Examples Find the number of terms in each of the following GPs. (b) 5, 20, 80, ……., 5120 (a) 2, 10, 50, ….., 1250 (a) a = 2, r = 5 (b) a = 5, r = 4 nth term = 2 x 5n- 1 = 1250 nth term = 5 x 4n- 1 = 5120 5n- 1 = 625 4n- 1 = 1024 trial and improvement trial and improvement 54 = 625 45 = 1024 n – 1 = 4 n – 1 = 5 n = 5 n = 6

  6. Examples Find the common ratio and the first term in these GPs. (a) the 2nd tem is 15 and the 5th term is 1875 (b) the 3rd term is 6 and the 7th term is 96 (a) 2nd term = ar = 15 [1] 5th term = ar4 = 1875 [2] [2] [1] = r3 = 125 giving r = 5 giving a = 3 Form [1] a x 5 = 15 (b) 3rd term = ar2 = 6 [1] 7th term = ar6 = 96 [2] [2] [1] = r4 = 16 giving r = 2 giving a = 1.5 Form [1] a x 4 = 6

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