Arithmetic and Geometric Progressions

May 8th, 2015
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An arithmetic progression is a list of numbers where the difference between successive numbers is constant. The terms in an arithmetic progression are usually denoted as u1 , u2 , u3 etc. where u1 is the initial term in the progression, u2 is the second term, and so on; un is the nth term

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ProgressionWorksheet 3.6 Arithmetic and Geometric ProgressionsSection 1 Arithmetic ProgressionAn arithmetic progression is a list of numbers where the dierence between successive numbers is constant. The terms in an arithmetic progression are usually denoted as u1 , u2 , u3 etc. where u1 is the initial term in the progression, u2 is the second term, and so on; un is the nth term. An example of an arithmetic progression is 2, 4, 6, 8, 10, 12, 14, . . . Since the dierence between successive terms is constant, we have u3 u 2 = u 2 u 1 and in general un+1 un = u2 u1 We will denote the dierence u2 u1 as d, which is a common notation. Example 1 : Given that 3,7 and 11 are the rst three terms in an arithmetic progression, what is d? 7 3 = 11 7 = 4 Then d = 4. That is, the common dierence between the terms is 4. If we know the rst term in an arithmetic progression , and the dierence between terms, then we can work out the nth term, i.e. we can work out what any term will be. The formula which tells us what the nth term in an arithmetic progression is un = a + (n 1) d where a is the rst term. Example 2 : If the rst 3 terms in an arithmetic progression are 3,7,11 then what is the 10th term? The rst term is a = 3, and the common dierence is d = 4. un = a + (n 1)d u10 = 3 + (10 1)4 = 3+94 = 39 1Example 3 : If the rst 3 terms in an arithmetic progression ar

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