Introduction to Growth by Addition
Rates of change can sometimes be expressed in terms of mathematical constants.
The simplest changeable quantities can be written as lists of numbers whose values repeatedly increase or decrease by a fixed amount. Here are some examples:
A = 1, 2, 3, 4, 5, 6
B = 0, −1, −2, −3, −4, −5
C = 2, 4, 6, 8
D = −5, −10, −15, −20
E = 4, 8, 12, 16, 20, 24, 28,. . .
F = 2, 0, −2, −4, −6, −8, −10,. . .
The first four of these sequences are finite. The last two are infinite, as indicated by the three dots following the last term in each case.
Arithmetic Progression
In each of the six sequences shown above, the values either increase ( A , C , and E ) or else they decrease ( B , D , and F ). In all six sequences, the “spacing” between numbers is constant throughout. Note:
- The values in A always increase by 1.
- The values in B always decrease by 1.
- The values in C always increase by 2.
- The values in D always decrease by 5.
- The values in E always increase by 4.
- The values in F always decrease by 2.
Each sequence has a starting point or first number. After that, succeeding numbers can be predicted by repeatedly adding a constant. If the added constant is positive, the sequence increases. If the added constant is negative, the sequence decreases.
Let s _{0} be the first number in a sequence S , and let c be a constant. Imagine that S can be written in this form:
S = s _{0} , ( s _{0} + c), ( s _{0} + 2c), ( s _{0} + 3c),. . .
for as far as the sequence happens to go. Such a sequence is called an arithmetic sequence or an arithmetic progression . In this context, the word “arithmetic” is pronounced “air-ith-MET-ick.”
The numbers s _{0} and c can be whole numbers, but that is not a requirement. They can be fractions such as 2/3 or −7/5. They can be irrational numbers such as the square root of 2. As long as the separation between any two adjacent terms in a sequence is the same, the sequence is an arithmetic progression. In fact, even if s _{0} and c are both equal to 0, the resulting sequence is an arithmetic progression.
Arithmetic Series
A series is, by definition, the sum of all the terms in a sequence. For any arithmetic sequence, the corresponding arithmetic series can be defined only if the sequence is finite. That means it must have a finite number of terms. For the above sequences A through F , let the corresponding series be called A _{+}
through F _{+} . Then:
A _{+} = 1 + 2 + 3 + 4 + 5 + 6 = 21 B _{+} = 0 + (−1) + (−2) + (−3) + (−4) + (−5) = −15 C _{+} = 2 + 4 + 6 + 8 = 20 D _{+} = (−5) + (−10) + (−15) + (−20) = −50
E _{+} is not defined
F _{+} is not defined
Arithmetic Interpolation and Extrapolation
Arithmetic Interpolation
When you see a long sequence of numbers, you should be able to tell without much trouble whether or not it’s an arithmetic sequence. If it isn’t immediately obvious, you can conduct a test: subtract each number from the one after it. If all the differences are the same, then the sequence is an arithmetic sequence.
Imagine that you see a long sequence of numbers, and some of the intermediate values are missing. An example of such a situation is shown in Table 14–1. It’s not too hard to figure out what the missing values are, once you realize that this is an arithmetic sequence in which each term has a value that is 3 larger than the term preceding it. The 4th term has a value of 14, and the 9th term has a value of 29. The process of filling in missing intermediate values in a sequence is a form of interpolation . We might call the process of filling in Table 14–1 arithmetic interpolation .
Table 14–1 A sequence with some intermediate values missing. They can be filled in by interpolation.
Position: |
1st |
2nd |
3rd |
4th |
5th |
6th |
7th |
8th |
9th |
10th |
11th |
Value: |
5 |
8 |
11 |
? |
17 |
20 |
23 |
26 |
? |
32 |
35 |
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