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11 SL notes 2015-16 Unit 1: Algebra Lesson 3: Arithmetic Series A series is the sum of a sequence. An arithmetic series is the sum of an arithmetic sequence. One of the most famous examples is told in an historical anecdote. While he was still in elementary school, Carl Freidrich Gauss 1777-1855, (who later became a very famous mathematician) was apparently asked by his teacher to find the sum of the numbers from 1 to 100. The teacher was very surprised when young Carl came back with the correct answer in a matter of seconds. You too can learn this technique. Let 𝑆𝑛 mean the sum of the first n terms of an arithmetic series, so in this case 𝑆100 = 1 + 2 + 3 + 4 + …+97 + 98 + 99 + 100 We can then rewrite the sequence in reverse order: 𝑆100 = 100 + 99 + 98 + 97 +…+ 4 + 3 + 2 + 1 Adding the two together gives 𝑆100 = 1 + 2 + 3 + 4 + …+97 + 98 + 99 + 100 𝑆100 = 100 + 99 + 98 + 97 +…+ 4 + 3 + 2 + 1 2𝑆100 = 101 + 101 + 101 + 101 +…+ 101 + 101 + 101 + 101, which is 100 terms, each of which is equal to the first term plus the last term. To find 𝑆100 this sum must be divided by two 100 Hence 𝑆100 = 2 (1 + 100) =5050. 𝑛 The general form is 𝑆𝑛 = 2 (𝑢1 + 𝑢𝑛 ). To use this formula as it stands one must know the first term, the last term and the number of terms, which information is not always available. We can express the last term using 𝑢1 and 𝑑, the common difference. 𝑛 𝑆𝑛 = 2 (𝑢1 + 𝑢𝑛 ), substitute 𝑢𝑛 = 𝑢1 + (𝑛 − 1)𝑑 𝑛 𝑆𝑛 = (𝑢1 + {𝑢1 + (𝑛 − 1)𝑑}) 2 𝑛 = 2 (2𝑢1 + (𝑛 − 1)𝑑) Check that this formula still works for the example above, them sum of the first one hundred counting numbers: 100 𝑆100 = 2 (2(1) + (100 − 1)1) = 50(101) = 5050, just as before. You can also check that the formula is the sum of the terms each expressed in terms of 𝑢1 and 𝑑: 𝑆𝑛 = 𝑢1 + (𝑢1 + 𝑑) + (𝑢1 + 2𝑑) + (𝑢1 + 3𝑑) + …+(𝑢1 + (𝑛 − 1)𝑑) 𝑛 𝑆𝑛 = 2 (2𝑢1 + (𝑛 − 1)𝑑) is the sum of an arithmetic series. It is a quadratic in n. 𝑆𝑛 is a quadratic function in n ⟺ 𝑢𝑛 is an arithmetic sequence. N.B. the sum to infinity of an arithmetic series is always plus or minus infinity. 11 SL notes 2015-16 e.g. For the arithmetic sequence 1, 5, 9, 13, … find 𝑆20 𝑢1 = 1, 𝑑 = 4 𝑛 𝑆𝑛 = {2𝑢1 + (𝑛 − 1)𝑑} 2 20 𝑆20 = {2(1) + (20 − 1)4} 2 =780 e.g. Find 𝑆20 for the sequence which has 𝑢5 = 19 and 𝑢7 = 27. 𝑢7 − 𝑢5 = 2𝑑 = 8, so d=4. Using 𝑢5 = 19, 19 = 𝑢1 + (5 − 1)4 Hence 𝑢1 = 3 𝑛 𝑆𝑛 = {2𝑢1 + (𝑛 − 1)𝑑} 2 20 𝑆20 = {2(3) + (20 − 1)4} 2 =820 e.g. In an arithmetic series, 𝑆20 = 710 and 𝑑 = 3. Find 𝑢1 , and hence find 𝑢25 and 𝑆25 . 𝑛 𝑆𝑛 = {2𝑢1 + (𝑛 − 1)𝑑} 2 20 𝑆20 = {2(𝑢1 ) + (20 − 1)3} 2 710=10{2(𝑢1 )+57} 𝑢1 = 710 −57 10 2 =7 𝑢25 = 7 + (25 − 1)3 = 79 𝑆25 = 25 {7 + 79} = 1075 2 e.g. Prove that the series whose sum for the first n terms is 𝑛2 must be an arithmetic series. 𝑛 𝑆𝑛 = {2𝑢1 + (𝑛 − 1)𝑑} 2 𝑛 2 𝑛 = {2𝑢1 + (𝑛 − 1)𝑑} 2 𝑑 𝑛 = 𝑢1 + (𝑛 − 1)( 2) which is a term in an arithmetic sequence. 11 SL notes 2015-16 e.g. In an arithmetic progression 𝑢1 = 5, 𝑑 = 3, and 𝑆𝑛 = 55. Find n, and hence write down all the terms that add to 55. N.B. Sigma notation. Σ is the capital Greek letter sigma, and it is also used to indicate a sum. e.g. 𝑛 ∑(𝑘 + 1) 𝑘=1 means the sum of the first n terms of the series 2+3+4+5+6… e.g. Evaluate 10 ∑(𝑘 − 3) 𝑘=1 Now try Ex 6F, page 169, focus on questions 4 to 12. You may come to M106 for study hall on Thursday P1&2, but you need to have tried the questions before you come.