22 ## SIGMA NOTATION FOR SUMSThe sum of consecutive numbers
For example: This means that we are to repeatedly add In other words, we are to repeatedly add
Problem 1. Write the following sums. To see the answer, pass your mouse over the colored area.
Example 1. Upper index
That is how to use sigma notation to indicate the sum of Problem 2. Write the following sum.
Here, the sum begins with
When the signs alternate, positive and negative in that way, we call that an alternating series. What causes the signs to alternate is (−1) See Lesson 13 of Algebra, Example 1 and Problem 7. Problem 3. Write the following sum.
Example 3. Use sigma notation to indicate this sum: 2 + 4 + 6 + 8 + . . . + 100.
2 Problem 4. Use sigma notation to indicate these sums.
Three theorems
A factor that does not depend on the index
A sum that consists of terms may be split into a sum over each term.
If the argument of the summation is a constant, that is, does not depend on the index
The sum of consecutive numbers In the Appendix to Arithmetic, we saw that the sum of consecutive numbers -- a triangular number -- is given by this formula: ½ where Example 4. Cite the theorems of summation to prove:
Remainder classes modulo Upon division of a number by 3, for example, the Here is the remainder class with remainder 0: 3 6 9 12 15 18 . . . Algebraically, these are the number 3 We say that these are the numbers congruent to 0 modulo 3. Here is the remainder class with remainder 1: 1 4 7 10 13 . . . These are the numbers 3 These are the numbers congruent to 1 modulo 3. Finally, here is the remainder class with remainder 2: 2 5 8 11 14 17 . . . These are the numbers 3 These are the numbers congruent to 2 (or to −1) modulo 3. Those are the three remainder classes modulo 3, that is, upon division by 3. Problem 5. a) Upon division by 5, what are the possible remainders? 0, 1, 2, 3, 4. b) Write the first four numbers of each remainder class. 5 10 15 20 1 6 11 16 2 7 12 17 3 8 13 18 4 9 14 19 c) Indicate each remainder class algebraically, and let each one begin
5
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5 An arithmetic series An arithmetic series is a sum in which each term is generated from the previous term by adding the same number. That number is called the constant difference. 2 + 5 + 8 + 11 + 14 + . . . + 32. The sum of consecutive numbers in a remainder class is an arithmetic series. 3 is the constant difference. It is the modulo of the remainder class. Example 5. Prove: 2 + 5 + 8 + 11 + 14 + . . . + (3
Here is the proof:
Problem 6. Prove: That is, 1 + 3 + 5 + . . . + (2
That is Problem 3 in Topic 27, Mathematical induction. And in the Appendix to Arithmetic, we explain directly why that is true. Problem 7. Prove: 1 + 5 + 9 + . . . + (4
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