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Repeated Addition
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Graphing an arithmetic sequence When we plot the values of an arithmetic sequence as a function of the term number in rectangular coordinates, we get a set of discrete points We can depict the term number along the horizontal axis going toward the right, so the term number plays the role of the independent variable We can plot the term value along the vertical axis, so it plays the role of the dependent variable Figure 19-1 illustrates two arithmetic sequences as they appear when graphed in this way (The dashed lines connect the dots, but they aren t actually parts of the sequences) One sequence is increasing, and the dashed line connecting this set of points ramps upward as we go toward the right Because this sequence is finite, the dashed line ends at (6,6) The other sequence is decreasing, and its dashed line ramps downward as we go toward the right This sequence is infinite, as shown by the ellipsis at the end of the string of numbers, and also by the arrow at the right-hand end of the dashed line When any arithmetic sequence is graphed according to the scheme shown in Fig 19-1, its points lie along a straight line The slope m of the line depends on whether the sequence increases ( positive slope) or decreases (negative slope) In fact, m is equal to the constant c in the general arithmetic series form:
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S = s0, (s0 + c), (s0 + 2c), (s0 + 3c), regardless of how many terms the sequence contains
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Rectangular-coordinate plots of two arithmetic sequences
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Sequences, Series, and Limits
An example Suppose that in an infinite sequence S, we have s0 = 5 and c = 3 The first 10 terms are
s0 = 5 s1 = s0 + 3 = 5 + 3 = 8 s2 = s1 + 3 = 8 + 3 = 11 s3 = s2 + 3 = 11 + 3 = 14 s4 = s3 + 3 = 14 + 3 = 17 s5 = s4 + 3 = 17 + 3 = 20 s6 = s5 + 3 = 20 + 3 = 23 s7 = s6 + 3 = 23 + 3 = 26 s8 = s7 + 3 = 26 + 3 = 29 s9 = s8 + 3 = 29 + 3 = 32 Therefore S = 5, 8, 11, 14, 17, 20, 23, 26, 29, 32,
Another example Consider the following sequence T Someone asks, Is this an arithmetic sequence If so, what are the values t0 (the starting value) and ct (the constant of change)
T = 2, 4, 8, 16, 32, 64, 128, 256, 512, In this case, T is not an arithmetic sequence The numbers do not increase at a steady rate There is a pattern, however Each number in the sequence is twice as large as the number before it
Still another example Consider the following sequence U Someone asks, Is this an arithmetic sequence If so, what are the values u0 (the starting value) and cu (the constant of change)
U = 100, 65, 30, 5, 40, 75, 110, This is an arithmetic sequence, at least for the numbers shown (the first seven terms) In this case, s0 = 100 and cu = 35, so we can generate the following list: s0 = 100 u1 = u0 + ( 35) = 100 35 = 65 u2 = u1 + ( 35) = 65 35 = 30 u3 = u2 + ( 35) = 30 35 = 5 u4 = u3 + ( 35) = 5 35 = 40 u5 = u4 + ( 35) = 40 35 = 75 u6 = u5 + ( 35) = 75 35 = 110
Repeated Addition
Are you confused
You ask, What happens if we start a sequence with a fixed number and then alternately add and subtract a constant Is the result an arithmetic sequence Here s an example: V = 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, In this case, the first term, v0, is equal to 1/2 We might say that the constant, cv, is equal to 1, but we alternately add and subtract it to generate the terms This is a definable sequence, but it s not an arithmetic sequence In order to generate a true arithmetic sequence, we must repeatedly add the constant, whether it s positive, negative, or 0 When the constant is positive, the terms steadily increase When the constant is negative, the terms steadily decrease Arithmetic sequences never alternate as V does
Here s a challenge!
When we have a sequence and we start to add up its numbers, we get another sequence of numbers representing the sums These sums are called partial sums List the first five partial sums of the following sequences: S = 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, T = 2, 4, 8, 16, 32, 64, 128, 256, 512, U = 100, 65, 30, 5, 40, 75, 110, V = 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2,
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