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ARITHMETIC SEQUENCES

ARITHMETIC SEQUENCES. Consider this sequence. If we subtract the second term from the first we find that the terms increase by four. 7, 11, 15, 19, …….. . 4. 4. 4. Because this sequence has a common difference between consecutive terms of 4 it is an arithmetic sequences .

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ARITHMETIC SEQUENCES

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  1. ARITHMETIC SEQUENCES

  2. Consider this sequence. If we subtract the second term from the first we find that the terms increase by four. 7, 11, 15, 19, …….. 4 4 4 Because this sequence has a common difference between consecutive terms of 4 it is an arithmetic sequences. This could be called a “linear sequence” because we could use the same techniques of finding the equation of lines to help us find the nth term or the pattern for this sequence.

  3. 7, 11, 15, 19, …….. n: 1 2 3 4 These can be written as points. (1, 7), (2, 11) and so on… We know that the slope is the common difference but we could find it using the slope formula too. Using y – y1 = m (x – x1) and any point from the sequence we can find the pattern (formula) of an arithmetic sequence. y – 7 = 4 (x – 1) which becomes y = 4x + 3 or an = 4n + 3

  4. Find the formula for the nth term of the arithmetic sequence whose common difference is 3 and whose first term is 2. What info are they giving us? What info do we need to find the equation of a “line”? The “first term is 2” translates into the point (1, 2) y – 2 = 3 (x – 1) which becomes y = 3x -1 an = 3n - 1

  5. The fourth term of an arithmetic sequence is 20 and the 13th term is 65. Write the first 5 terms of this sequence. What info are they giving us? What do we wish they would have given us? What info do we need to find the equation of a “line”? Gave us two points (4, 20) and (13, 65). We can use those to find the slope of 5. We can now find the formula: y – 20 = 5 (x – 4) which an = 5n 5 10 15 20 25 _____ _____ _____ _____ _____ 1 2 3 4 5

  6. Finding the SUM of a FINITE ARITHMETIC SEQUENCE Time for a shortcut! Formula: Sn = Where n is the upper index or the total number of terms in the sequence, a1 is the first term and an is the last term. 5 10 15 20 25 Sn = 75

  7. Finding the SUM of a FINITE ARITHMETIC SEQUENCE We need to find the first and last terms. 11(1) – 6 = 5 11(150) – 6 = 1644 Sn = 123,675

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