Arithmetic and Geometric Sequences, Study Guides, Projects, Research of Calculus for Engineers

Download Arithmetic and Geometric Sequences and more Study Guides, Projects, Research Calculus for Engineers in PDF only on Docsity! Arithmetic and Geometric Sequences A sequence is a list of numbers or objects, called terms, in a certain order. In an arithmetic sequence, the difference between one term and the next is always the same. This difference is called a common difference. The common difference is added to each term to get the next term. 2, 5, 8, 11, 14, … This is an increasing arithmetic sequence with a common difference of 3. 32, 26, 20, 14, 8, … This is a decreasing arithmetic sequence with a common difference of –6. Example: What are the next three terms in the sequence? 1, 5, 9, 13, … I can see that this is an arithmetic sequence with a common difference of 4. To get the next three terms, add 4 to 13 which equals 17, the next term in the sequence. Then add 4 to 17 to get the next term to get 21, etc. So the next three terms are 17, 21, and 25. Use the following formula to find any term of an arithmetic sequence. 1 ( 1)na a n d= + − an = the term in the sequence you are trying to find (n represents the desired term number) a1 = the first term in the sequence d = the common difference Example: What is the 10th term of the following sequence? 1, 5, 9, 13, … 10 1 (10 1)4 1 9 4 1 36 37a + − = + ⋅ = + == So the 10th term of this sequence is 37. Example: What is the 12th term of the following sequence? 34, 31, 28, 25, 22, … 12 34 (12 1)( 3) 34 11( 3) 34 ( 33) 1a + − − = + − = + − == The 12th term of this sequence is 1. Practice: 1. Find the next three terms: 3, 10, 17, 24, 31, _____, _____, _____ 2. Find the 25th term: 53, 50, 47, 44, 41, … ________ 3. Find the 20th term: 25, 40, 55, 70, 85, … ________ 4. Find the 75th term: 88, 81, 74, 67, 60, … ________ A geometric sequence is a sequence of numbers where the ratio of consecutive terms is constant. This ratio is called the common ratio (r). Sometimes the terms of a geometric sequence get so large that you may need to express the terms in scientific notation rounded to the nearest tenth. 2, 6, 18, 54, … This is an increasing geometric sequence with a common ratio of 3. 1,000, 200, 40, 8, … This is a decreasing geometric sequence with a common ratio or 0.2 or 5 1 . Example: What are the next three terms of the following sequence? 4, 20, 100, 500, … 500,625500,12 500,1252500 25005500 =⋅ =⋅ =⋅ The next three terms are 2,500, 12,500, and 62,500. Explicit sequences also have a formula for finding any term in a sequence. 1 ( 1) n na a r −= an = the term in the sequence you are trying to find (n represents the desired term number) a1 = the first term in the sequence r = the common ratio Example: Find the 7th term in the following sequence: 6, 18, 54, 162, … Finding the common ratio can be harder than finding the common difference. One way to find it is the divide each term by the term before it. 3618 =÷ , 31854 =÷ , 354162 =÷ So the common ratio is 3. (7 1) 6 7 6 3 6 3 6 729 4,374a −⋅ ⋅ ⋅ == = = So the 7th term of the sequence is 4, 374. Example: Find the 8th term in the following sequence: 96, 48, 24, 12, 6, … To find the common ratio, divide each term by the one before it. 2 19648 =÷ , 2 14824 =÷ , 2 12412 =÷ The common ratio is 2 1 . 3. Use your formula from question 2c to find the values of a7 and a20. 4. For the following geometric sequences, find a and r and state the formula for the general term. a) 1, 3, 9, 27, ... b) 12, 6, 3, 1.5, ... c) 9, -3, 1, ... 5. Use your formula from question 4c) to find the values of the a4 and a12 6. Find the number of terms in the following arithmetic sequences. Hint: you will need to find the formula for tn first! a) 2, 5, 8, ..... , 299 b) 9, 5, 1, ..... - 251. Answers: 1a) arithmetic d = 6 b) neither c) geometric r = 7 d) geometric r = 0.5 or r = ½ e) arithmetic d = 15 f) geometric r = -3 2a) a = -10; d=6; tn =6n-16 b) a = 10; d=-2; tn=-2n+12 c) a = 36; d=-5; tn= - 5n+41 3. t7=6; t20 = -59 4. a) a = 1; r = 3; tn = 1(3)n –1 b) a = 12; r = ;a c) a = 9; r = - 3; tn = 9(-3)n –1 5. t4 = -243 t12 = -177147 6. a) tn = 3n-1; n = 100 b) tn = -4n+13; n=66 General formula for a geometric series: 1) Find the designated sum of the arithmetic series a) 𝑆14 of 3+7+11+15+⋯ b) 𝑆11 of −13−11−9−7−⋯ c) 𝑆9 of 22+20+18+16+⋯ d) 𝑆35 of −2−5−8−11−⋯ 2) Determine the sum of each arithmetic series a) 6+13+20+⋯+69 b) 4+15+26+⋯+213 c) 5−8−21−⋯−190 d) 100+90+80+⋯−100 Arithmetic and Geometric Series – Worksheet General formula for an arithmetic series: 3) Find the designated sum of the geometric series a) 𝑆7 of 4+8+16+32+⋯ b) 𝑆13 of 1−6+36−216+⋯ c) 𝑆17 of 486+162+54+18+⋯ 4) Determine 𝑆𝑛 for each geometric series d) 𝑆6 of 3+15+75+375+⋯ a) 𝑎=6, 𝑟=2, 𝑛=9 b) 𝑓 1 =2, 𝑟=−2, 𝑛=12 c) 𝑓 1 =729, 𝑟=−3, 𝑛=15 d) 𝑓 1 =2700, 𝑟=10, 𝑛=8 5) If the first term of an arithmetic series is 2, the last term is 20, and the increase constant is +2 … a) Determine the number of terms in the series b) Determine the sum of all the terms in the series 6) A geometric series has a sum of 1365. Each term increases by a factor of 4. If there are 6 terms, find the value of the first term.