Math Sequences and Series Worksheets

Pre-Calculus Sequences & Series Worksheet Write the first five terms of the sequence. Determine whether or not the sequence is arithmetic. If it is, find the common difference. 1 1. 𝑎𝑛 = 8 + 13𝑛 2. 𝑎𝑛 = 𝑛+1 3. 𝑎𝑛 = 2𝑛 + 𝑛 Find the nth term of the sequence, then find the 20th term. 4. 𝑎1 = 2 and 𝑑 = 3 5. −6, −4, −2, …. 6. 𝑎1 = 0 and 𝑑 = Find the nth term of the sequence. 8. 𝑎1 = −4 and 𝑎5 = 16 9. 𝑎3 = 94 and 𝑎6 = 85 2 3 7. 2 1 −4 , , ,… 5 15 15 10. 𝑎5 = 190 and 𝑎10 = 115 Find the nth term of the sequence. 11. 𝑎6 = −38 and 𝑎11 = −73 12. 𝑎3 = 19 and 𝑎15 = −1.7 13. 𝑎5 = 16 and 𝑎14 = 38.5 Find the indicated nth partial sum (Sn) of the arithmetic sequence. 14. 8, 20, 32, 44, … 𝑛 = 10 15. 𝑎1 = −6, 𝑑 = 4, 𝑛 = 50 16. 100 + 105 + 110 + ⋯ + 220 17. 0.5 + 1.3 + 2.1 + … + 70.1 18. 𝑎1 = 3, 𝑑 = 2, 𝑛 = 12 19. 𝑎1 = 100, 𝑑 = −5, 𝑛 = 8 20. 𝑎2 = 8, 𝑎5 = 9.5, 𝑛 = 12 21. −3 + ( 2 ) + 0 + ⋯ + 30 −3 Find the sums of the following arithmetic series in summation notation. 50 22. ∑ 100 23. ∑ 𝑛 𝑛=1 𝑛 = 51 250 30 25. ∑ (600 − 𝑛) 𝑛 = 100 26. ∑ 𝑛 = 11 500 24. ∑ 2𝑛 (𝑛 + 6) 𝑛 = 75 𝑛 − ∑ 10 17 𝑛 27. ∑ 𝑛=1 28. How many terms of the arithmetic sequence −2, 3, 8,… must be added to get 1573? 29. How many terms of the arithmetic sequence 15, 12, 9,…must be added to get −39? 30. How many terms of the arithmetic sequence −1, 2, 5,… must be added to get 609? 𝑛=2 2𝑛 − ∑ 10 𝑛 𝑛=5 Answers 1 1 1 1 1 , , , , ; 2 3 4 5 6 1. 21, 34, 47, 60, 73; Arithmetic; 𝑑 = 13 2. 3. 3, 6, 11, 20, 37; Not Arithmetic 4. 𝑎𝑛 = 3𝑛 − 1; 𝑎20 = 59 2 2 6. 𝑎𝑛 = 3 𝑛 − 3; 𝑎20 = 38 3 1 Not Arithmetic 11 5. 𝑎𝑛 = 2𝑛 − 8 ; 𝑎20 = 32 89 7. 𝑎𝑛 = − 3 𝑛 + 15; 𝑎20 = − 15 8. 𝑎𝑛 = 5𝑛 − 9 9. 𝑎𝑛 = −3𝑛 + 103 10. 𝑎𝑛 = −15𝑛 + 265 11. 𝑎𝑛 = −7𝑛 + 4 12. 𝑎𝑛 = −1.725𝑛 + 24.175 13. 𝑎𝑛 = 2.5𝑛 + 3.5 14. 𝑆10 = 620 15. 𝑆50 = 4,600 16. 𝑆25 = 4,000 17. 𝑆88 = 3,106.4 18. 𝑆12 = 168 19. 𝑆8 = 660 20. 𝑆12 = 123 22. 𝑆50 = 1,275 23. 𝑆50 = 7,550 24. 𝑆426 = 125,031 25. 𝑆151 = 64,175 26. 𝑆 = 355 27. 𝑆 = 259 28. 𝑛 = 26 29. 𝑛 = 13 21. 𝑆23 = 621 2 30. 𝑛 = 21 Pre-Calculus Sequences & Series Worksheet The nth term of a sequence is given. Find the first five terms of the sequence. 1. 𝑎𝑛 = 3(−4)𝑛−1 2. 𝑎𝑛 = 3𝑛−1 Find the nth term or the geometric sequence with giver first term a and a common ratio r. What is the fourth term? 3. 𝑎 = −6, 𝑟 = 3 4. 𝑎 = √3, 𝑟 = √3 Determine if the sequence is geometric. If it is geometric, find the common ratio. 5. 2, 6, 18, 36 … 6. 27, −9, 3, −1 … 7. 1 1 1 1 , , , … 2 4 6 8 Find the first five terms of the sequence and determine if it is geometric. If it is geometric, find the common ratio and express the nth term of the sequence in the standard form 𝒂𝒏 = 𝒂𝒓𝒏−𝟏 8. 𝑎𝑛 = 4 + 3𝑛 9. 𝑎𝑛 = (−1)𝑛 2𝑛 10. 𝑎𝑛 = 𝑛𝑛 Determine the common ratio, the fifth term, and the nth term of the geometric sequence. 11. 7, 14 28 56 , 9 , 27 … 3 12. 1, √2, 2, 2√2 … 4 14. The first term of a geometric sequence is 3, and the third term is 3. Find the fifth term. 3 1 1 13. −8, −2, − 2, − 8 … 15. The common ratio in a geometric sequence is 2, and the fifth term is 1. Find the first three terms. For the following problems, find the sum of the infinite geometric series, if possible. 1 1 1 16. 1 − 2 + 4 − 8 + ⋯ 17. 2 4 + 25 5 8 + 125 + ⋯ 3 3 3 18. 3 − 2 + 4 − 8 + ⋯ Express the repeating decimal as a fraction. ̅̅̅̅ 19. 0.253 20. 0.123123123123 … Answers 1. 3, −12, 48, −192, 768 2. 1, 3, 9, 27, 81 3. 𝑎𝑛 = −6(3)𝑛−1 , 𝑎4 = −162 4. 𝑎𝑛 = √3(√3) 5. Not geometric 6. 𝑟 = − 3 7. Not geometric 8. 7, 13, 31, 85, 247; Not geometric 9. −2, 4, −8, 16, −32; 𝑟 = −2; 𝑎𝑛 = −2(−2)𝑛−1 10. 1, 4, 27, 256, 3125; Not geometric 2 ; 3 𝑎5 = 13. 𝑟 = 1 ; 4 𝑎5 = − 16 8 4 , , 81 27 9 2 17. 3 19. 251 990 , 𝑎4 = 9 1 11. 𝑟 = 15. 𝑛−1 112 81 2 𝑛−1 ; 𝑎𝑛 = 7 (3) 1 32 1 𝑛−1 ; 𝑎𝑛 = −8 (4) 𝑛−1 12. 𝑟 = √2; 𝑎5 = 4 ; 𝑎𝑛 = 1(√2) 14. 𝑎5 = 16. 2 3 18. 2 20. 123 999 16 27