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Definitions: A progression of numbers is called a sequence. E.g., 2, 4, 6, 8,…, 100. The sequence can be finite if it ends or infinite if it is unending. In our example, the sequence ends with the 50th term, 100, and is therefore a finite sequence. Sequences and Series Level 1 Book 1 2 A term of a sequence (or series) is an individual number in the progression. In our example above, the 3rd term is 6. A Series is the sum of a progression of numbers. E.g., 2+4+6+8+…+100. Often we are interested in the particular number that this sum equals, in our example we can most easily find the sum if we rearrange the addition: (2+100)+(4+98)+ … + (50+52). In this way we see that the sum is half the number of terms, 25, times the sum of the first and last term, 102= 2550. Differences in a sequence are the numbers obtained by subtracting a term from its successive term. In our example, the differences are: 4-2 = 2 6-4 = 2 etc. An Arithmetic Sequence is a sequence in which all the differences are the same. Our example 2, 4, 6,…, 100 is an arithmetic sequence because the differences all equal 2. A second example, 1, 3, 6, 10,…, 45 is not an arithmetic sequence because its differences for the sequence 1, 2, 3,…, 9 and hence the differences aren’t all the same. An Arithmetic Series is the sum of an arithmetic sequence. In an arithmetic sequence (or series,) the constant difference of consecutive terms is called the Delta. The General Term of a sequence (or series) is a kind of basket that represents all of the terms of the sequence at once. In our example 2, 4, 6, … the general term is {2x}. When x=3, we see that 2x=6 is the third term. The general term makes it easy to predict that the 100th term is 200 or that the 1034th term is 2068. 3 Sequences and Series Level 1 Book 1 Technique for Arithmetic Series 1. Find the delta or common difference of all the terms. This identifies the times table that your series is a variation on, and hence tells you what to multiply x by in the general term. In the example 3+5+7+9+ …+97 this delta would be 2 telling us that this sequence is a variation on the 2-times table series: 2+4+6+8+…+96 2. Find what term zero of the sequence would be. In our example above, term zero is what would come before the 3 in 3+5+7+9+…+97. This would be 1. This is what must be added to the multiple of x (2) found in step 1. 3. Put the above two parts together and you get the general term: {2x+1}. 4. Now you are in a position to use the general term to make predictions. For instance, the 1000th term is 2001. We can also find out what term number is 97 by solving the equation 2x+1=97 to find that this is term number 48. 5. Now rearrange the terms: (3+97)+(5+95)+(7+93)+…+(49+51) where you can easily see that our sum equals 100 * 24 (24=half the number of terms— needed since we pair them up.) Sum=2400. Find the sum of the series: 11+15+19+23+…+207 Delta: 4 Term 0: 7 General Term: 4x + 7 What term number is 207: 50 Find the sum of the end terms: Find the sum of the series: 218 = 5450 Find the sum of the series: 4+11+18+25+…+137 Delta: 7 Term 0: -3 General Term: 7x - 3 What term number is 137: 20 218 25 * Sequences and Series Level 1 Book 1 Find the sum of the end terms: Find the sum of the series: 141 = 1410 4 141 10 * 5 Sequences and Series Level 1 Book 1 Find the sum of 10 terms of the series: 3+9+15+21+… Delta: Term 0: General Term: Term 10: Find the sum of the end terms: Find the sum of the series: Find the sum of the series: 20+30+40+50+…+120 Delta: Term 0: General Term: What term number is 120: Find the sum of the end terms: Find the sum of the series: Find the sum of 70 terms of the series: 12+16+20+24+… Delta: Term 0: General Term: Term 70: Sequences and Series Level 1 Book 1 Find the sum of the end terms: Find the sum of the series: Find sum of 10 terms of the series: 7+8+9+10+… Delta: Term 0: General Term: Term 10: Find the sum of the end terms: Find the sum of the series: Find sum of 60 terms of the sequence 9+12+15+18+… 6 7 Sequences and Series Level 1 Book 1 Find sum of the series 12+19+26+33+…+705 Sum the series 1+9+17+25+…+393 Sum 16 terms of the series: 19+27+35+43+… Sequences and Series Level 1 Book 1 Sum the series: 1+2+3+…+1000 8 9 Sequences and Series Level 1 Book 1 New Skill Introduced—Summation Notation Mathematicians use the capital Greek letter to indicate summation. Hence, 100 2n 2 4 6 ... 200 n 1 In the following examples, expand the summation to find the series you are working with. Then sum the series as you have done before. Find 100 (2n 1) n 1 Actual Series to be summed: 1+3+5+…+199 50 Find (3n 2) n 1 Actual Series to be summed: Sequences and Series Level 1 Book 1 63 (5n 8) n 1 Find 40 (8n 7) n 1 Find 30 (3n 1) n 1 Find 10 11 Sequences and Series Level 1 Book 1 11 ( 4 n 6) n 1 Find 100 (10n 4) n 1 Find 20 (4n 5) n 1 Sequences and Series Level 1 Book 1 Find 63 (5n 2) n 1 Find 30 ( 4 n 9) n 1 Find 12 13 Sequences and Series Level 1 Book 1 Find Answers Page No. 3 4 5 Exampl e No. 1 2 1 2 3 1 2 3 Delta 4 7 6 10 4 1 3 7 Term 0 7 -3 -3 10 8 6 6 5 General Term or Expanded Series 4x+7 7x-3 6x-3 10x+10 4x+8 x+6 3x+6 7x+5 nth Term n=50 n=20 57 n=11 288 16 186 n=100 Sum of End Terms 218 141 60 140 300 23 195 717 Series Sum 5450 1410 300 770 10500 115 5850 35850 -7 11 0 -1 2 8 -7 1 6 4 8x-7 8x+11 x 1+3+5+… 5+8+11+… 13+18+23+… 1+9+17+… 4+7+10+… 10+14+18+… 114+24+34+… n=50 139 n=1000 199 152 323 313 91 50 1004 384 158 1001 200 157 336 314 95 60 1018 9850 1264 500500 10000 3925 10584 6280 1425 330 50900 60 ( 6 n 9) n 1 6 7 8 9 1 2 3 1 2 1 2 3 1 2 8 8 1 2 3 5 8 3 4 10 Sequences and Series Level 1 Book 1 10 3 1 2 3 4 4 4 6 5 6 9 9 9+13+17+… 7+12+17+… 13+17+21+… 15+21+27+… 85 317 129 369 14 94 324 142 384 940 10206 2130 11520 Green Valley School 389 Pembroke Street Pembroke, NH 03275 (603) 485-8550