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Arithmetic sequences

Arithmetic sequences. An arithmetic sequence or arithmetic progression (AP) , is a sequence whose terms go up or down by constant steps i.e. there is a common difference. Examples:. (i) 5, 7, 9, 11, 13, ……. (ii) 88, 78, 68, 58, 48, …….

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Arithmetic sequences

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  1. Arithmetic sequences An arithmetic sequence or arithmetic progression (AP), is a sequence whose terms go up or down by constant steps i.e. there is a common difference. Examples: (i) 5, 7, 9, 11, 13, ……. (ii) 88, 78, 68, 58, 48, …….  The first term of an AP is denoted by a: u1 = a  The common difference is denoted by d: un+1 = un+ d  Formula for the nth term of AP is a + (n – 1)d  nth term: un = a + (n – 1)d ora + d(n – 1)

  2. Examples Decide which of the following sequences are arithmetic progressions. If the sequence is AP then write the common difference (d) and the next term. (a) 3, 6, 9, 12, 15, 18, ….. (b) 2, 4, 8, 16, 32, 64, … (c) 1, 0.5, 0, -0.5, -1.0, -1.5. ….. (d) m, m – n, m – 2n, m – 3n, m – 4n, … (a) yes: d = 3, Next term = 21 (b) no (c) yes: d = -0.5, Next term = -2. 0 (d) yes: d = -n, Next term = m – 5n

  3. Finding the nth term Write down the nth term for the following APs. (a) 2, 6, 10, 14, 18, ….. (b) 20, ,14, 8, 2, -4, …. (a) The first term is a = 2 The common difference d = 4 nth term: un = a + d(n – 1) = 2 + 4(n – 1) = 2 + 4n - 4 = 2n - 2 (b) The first term is a = 20 The common difference d = -6 nth term: un = a + d(n – 1) = 20 + -6(n – 1) = 20 - 6n + 6 = 26 – 6n

  4. Examples Write down an expression forthe nth term and find the 10th term, of the following arithmetic sequences. (a) 2.2, 2.5, 2.8, 3.1, …. (b) 3 + 2x, 3 + x, 3 , 3 – x, …. (a) a = 2.2 and d = 0.3 nth term: un = a + d(n – 1) = 2.2 + 0.3(n – 1) = 2.2 + 0.3n – 0.3 = 1.9 + 0.3n 10th term = 1.9 + 0.3 x 10 = 4. 9 (b) a = 3 + 2x and d = - x nth term: un = a + d(n – 1) = 3 + 2x – x(n -1) = 3 + 2x – xn + x = 3 + 3x – xn 10th term = 3 + 3x – 10x = 3 – 7x

  5. Examples Find the number of terms in the AP: 3, 7, 11, 15, 19, ……, 103 a = 3, d = 4, nth term = un = a + d(n – 1) 103 = 3 + 4(n – 1) 103 = 3 + 4n - 4 103 = 4n - 1 104 = 4n 26 = n So there are 26 terms in the AP.

  6. Examples Find the number of terms in each of the following APs. (b) -8, -6, , -4, …, 78 (a) 46, 42, 38, ….., -26 (a) a = 46, d= -4 (b) a = -8, d = 2 nth term = 46-4(n – 1) = -26 nth term = -8 + 2(n – 1) = 78 46 – 4n + 4 = - 26 - 8 + 2n - 2 = 78 50 – 4n = -26 2n – 10 = 78 -4n = -76 2n = 88 n = 19 n = 44

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