Arithmetic Series: Finding Sums of Consecutive Integers and Series Formulas, Exams of Algebra

Download Arithmetic Series: Finding Sums of Consecutive Integers and Series Formulas and more Exams Algebra in PDF only on Docsity! 868 Series and Combinations Lesson Arithmetic Series Chapter 13 13-1 BIG IDEA There are several ways to fi nd sums of the successive terms of an arithmetic sequence. Sums of Consecutive Integers There is a story the famous mathematician Carl Gauss often told about himself. When he was in third grade, his class misbehaved and the teacher gave the following problem as punishment: “Add the whole numbers from 1 to 100.” Gauss solved the problem in almost no time at all. His idea was the following. Let S be the desired sum. S = 1 + 2 + 3 + … + 98 + 99 + 100 Using the Commutative Property of Addition, the sum can be rewritten in reverse order. S = 100 + 99 + 98 + ... + 3 + 2 + 1 Now add corresponding terms in the equations above. The sums 1 + 100, 2 + 99, 3 + 98, ... all have the same value! So 2S = 101 + 101 + 101 + … + 101 + 101 + 101 . 100 terms Thus, 2S = 100 · 101 and S = 5050. Gauss wrote only the number 5050 on his slate, having done all the fi guring in his head. The teacher (who had hoped the problem would keep the students working for a long time) was quite irritated. However, partly as a result of this incident, the teacher did recognize that Gauss was extraordinary and gave him some advanced books to read. (You read about Gauss’s work in Lesson 11-6 and may recall that he proved the Fundamental Theorem of Algebra at age 18.) QY1 Mental Math Consider the arithmetic sequence defi ned by an = 3n - 12. a. Find a1, a2, and a3. b. Find a1 + a2 + a3. c. Find a101, a102, and a103. d. Find a101 + a102 + a103. ⎧      ⎨      ⎩⎧      ⎨      ⎩ QY1 Use Gauss’s method to add the integers from 1 to 40. Vocabulary series arithmetic series ∑, sigma ∑-notation, sigma notation, summation notation index variable, index SMP_SEAA_C13L01_868-875.indd 868 12/5/08 2:35:16 PM Arithmetic Series 869 Lesson 13-1 What Is an Arithmetic Series? Recall that an arithmetic or linear sequence is a sequence in which the difference between consecutive terms is constant. An arithmetic sequence has the form a1, a1 + d, a1 + 2d, ..., a1 + (n - 1)d, ... , where a1 is the fi rst term and d is the constant difference. For example, the odd integers from 1 to 999 form a fi nite arithmetic sequence with a1 = 1, n = 500, and d = 2. A series is an indicated sum of terms of a sequence. For example, for the sequence 1, 2, 3, a series is the indicated sum 1 + 2 + 3. The addends 1, 2, and 3 are the terms of the series. The value, or sum, of the series is 6. In general, the sum of the fi rst n terms of a series a is Sn = a1 + a2 + a3 + ... + an–2 + an–1 + an. If the terms of a series form an arithmetic sequence, the indicated sum of the terms is called an arithmetic series. If a is an arithmetic series with fi rst term a1 and constant difference d, you can fi nd a formula for the value Sn of the series by writing the series in two ways: Start with the fi rst term a1 and successively add the common difference d. Sn = a1 + (a1 + d) + (a1 + 2d) + ... + (a1 + (n - 1)d) Start with the last term an and successively subtract the common difference d. Sn = an + (an - d) + (an - 2d) + ... + (an - (n - 1)d) Now add corresponding pairs of terms of these two formulas, as Gauss did. Then each of the n pairs has the same sum, a1 + an. Sn + Sn = (a1 + an) + (a1 + an) + (a1 + an) + … + (a1 + an) n terms So 2Sn = n(a1 + an). Thus, Sn = n _ 2(a1 + an). This proves that if a1 + a2 + ... + an is an arithmetic series, then a formula for the value Sn of the series is Sn = n _ 2(a1 + an). QY2 Arithmetic series that involve the sum of consecutive integers from 1 to n lead to a special case of the above formula. In these situations, a1 = 1 and an = n, so the sum of the integers from 1 to n is n _ 2(1 + n), or n 2 + n ___ 2 . ⎧        ⎨        ⎩⎧        ⎨        ⎩ QY2 Use the formula for S n to fi nd the sum of the odd integers from 1 to 999. SMP_SEAA_C13L01_868-875.indd 869 12/5/08 2:35:21 PM 872 Series and Combinations Chapter 13 When ai = i, the sequence is the set of all positive integers in increasing order 1, 2, 3, 4, … . Then ∑ i = 1 n i = n _ 2(1 + n) = n(n + 1) ____ 2 . This is a ∑-notation version of Gauss’s sum. QY3 One advantage of ∑-notation is that you can substitute an expression for ai. For instance, suppose an = 2n, the sequence of even positive integers. Then, ∑ i = 1 6 ai = ∑ i = 1 6 (2i) = 2 · 1 + 2 · 2 + 2 · 3 + 2 · 4 + 2 · 5 + 2 · 6 = 2 + 4 + 6 + 8 + 10 + 12 = 42. The sum of the fi rst six positive even integers is 42. Example 3 Consider ∑ i = 1 500 a i , where a n = 4n + 6. a. Write the series without Σ-notation. b. Evaluate the sum. Solution a. Substitute the expression for a i from the explicit formula and use it to write out the terms of the series. ∑ i = 1 500 (4i + 6) = (4 · ? + 6) + (4 · ? + ? ) + ? + ... + ? = ? + ? + ? + ... + ? b. This is an arithmetic series. The fi rst term is ? . The constant difference is ? . There are ? terms in the series. Use the formula ∑ i = 1 n ai = n _ 2 (2a1 + (n - 1)d) to evaluate the series. ∑ i = 1 500 (4i + 6) = ? __ 2 (2 · ? + ( ? - 1) ? ) = ? QY3 Find ∑ i = 1 40 i . GUIDED SMP_SEAA_C13L01_868-875.indd 872 12/5/08 2:36:03 PM Arithmetic Series 873 Lesson 13-1 Most scientifi c calculators and CAS have commands to evaluate a series, but the commands and entry styles vary considerably. The entry for one CAS is shown at the right. QY4 Questions COVERING THE IDEAS In 1 and 2, tell whether what is given is an arithmetic sequence, an arithmetic series, or neither. 1. 26 + 29 + 32 + 35 2. 35, 32, 29, 26 3. What problem was Gauss given in third grade, and what is its answer? 4. Find the sum of the integers from 1 to 500. 5. Fill in the Blank The symbol ∑ is the upper-case Greek letter ? . 6. Fill in the Blank In ∑-notation, the variable under the ∑ sign is called the ? . 7. Consider the arithmetic sequence with fi rst term a1 and constant difference d. a. Write a formula for the nth term. b. Write a formula for the sum Sn of the fi rst n terms using ∑-notation. c. Write an equivalent formula to the one you wrote in Part b. 8. Consider the arithmetic series 8 + 13 + 18 + ... + 38. a. Write out all the terms of the series. How many terms are there? b. What is the sum of all the terms? c. Write this series using ∑-notation. 9. Refer to Example 3. Check your answer using the formula ∑ i = 1 n ai = n _ 2(a1 + an). 10. Multiple Choice ∑ i = 1 5 i2 = ? . A 1 + 4 + 9 + 16 + 25 B 52 C 1 + 2 + ... + 5 D none of these QY4 Check the solution to Example 3 on your calculator or CAS. SMP_SEAA_C13L01_868-875.indd 873 12/5/08 2:36:07 PM 874 Series and Combinations Chapter 13 11. Multiple Choice 7 + 14 + 21 + 28 + 35 + 42 + 49 + 56 + 63 = ? . A ∑ i = 7 63 i B ∑ i = 7 63 (7i) C ∑ i = 1 9 (7i) D none of these 12. In ∑ i = 100 300 (5i) , how many terms are added? (Be careful!) In 13 and 14, evaluate the sum. 13. ∑ i =1 25 (6i - 4) 14. ∑ n = –1 3 9 · 3n APPLYING THE MATHEMATICS 15. Finish this sentence: The sum of the fi rst n terms of an arithmetic sequence equals the average of the fi rst and last terms multiplied by ? . 16. The Jewish holiday Chanukah is celebrated by lighting candles in a menorah for eight nights. On the fi rst night, two candles are lit, one in the center and one on the right. The two candles are allowed to burn down completely. On the second night, three candles are lit (one in the center, and two others) and are allowed to burn down completely. On each successive night, one more candle is lit than the night before, and all are allowed to burn down completely. How many candles are needed for all eight nights? 17. Penny Banks decides to start saving money in a Holiday Club account. At the beginning of January she will deposit $100, and each month thereafter she will increase the deposit amount by $25. How much will Penny deposit during the year? 18. a. How many even integers are there from 50 to 100? b. Find the sum of the even integers from 50 to 100. 19. a. Write the sum of the squares of the integers from 1 to 100 in ∑-notation. b. Evaluate the sum in Part a. 20. a. Translate this statement into an algebraic formula using ∑-notation: The sum of the cubes of the integers from 1 to n is the square of the sum of the integers from 1 to n. b. Verify the statement in Part a when n = 8 21. Write the arithmetic mean of the n numbers a1, a2, a3, …, an using ∑-notation. SMP_SEAA_C13L01_868-875.indd 874 12/5/08 2:36:11 PM