SEQUENCES AND SERIES A sequence is a set of numbers in a defined order (a pattern). There are two kinds of sequences: arithmetic and geometric. An arithmetic sequence uses a number called d (difference) that is added to get from one number to the next in the sequence. For example: 2, 5, 8, 11,... is an arithmetic sequence with d = 3. A geometric sequence uses one number called r (ratio) that is multiplied to get from one number to the next. For example: 2, 6, 18, 54,... is a geometric sequence with r = 3. For both of these sequences, u1 is the first term, and n is the number of terms in the sequence. For example: For the arithmetic sequence: 3, 6, 9, 12, 15, 18, 21, 24, the first term u1 is 3 and the number of terms, n = 8. For example: For the geometric sequence: 10, 20, 40, 80, 160, the first term u1 is 10 and the number of terms, n = 5. Both arithmetic and geometric sequences can have negative numbers, fractions, and decimal numbers as difference or ratios. For example: The geometric sequence: -.5,.25, -.125,.0625,... Notice that since the ratio, r = -.5, the signs of the terms alternate between positive and negative. You will be responsible for finding the first term, the last term, and solve problems using sequences. These can be solved using the formulae in the Information Booklet Topic 2. Series is the sum of either an arithmetic or geometric sequence. The formulae to find these sums are in the Information Booklet Topic 2.
EXAMPLES Find the difference or ratio for each: 1. 4, 7, 10, 13, 16,... Ans. Difference is 3 2. 1, 2, 4, 8, 16,... Ans. Ratio is 2 3. 10, 8, 6, 4, 2,... Ans. Difference is -2 4. -12,-8,-4, 0, 4,... Ans. Difference is 4 5. 5.3, 5.7, 6.1, 6.5, 6.9,... Ans. Difference is.4 6. 6, -12, 24, -48,... Ans. Ratio is -2 Find un, the last term (nth term) of the arithmetic sequence: 1. u1 = 1, d = 3, n = 10 Ans. 28 2. u1 = 15, d = 4, n = 25 Ans. 111 Write the first 5 terms of the geometric sequence: 1. u1 = 4, r =.5 Ans. 4, 2,.5,.25,.125 2. u1 = 5, r = 3 Ans. 5, 15, 45, 135, 405 Find the nth term, the last term: 1. u1 = 2, r = 5, n = 6 Ans. 6250 2. u1 = 4, r =.5, n = 7 Ans..0625
SERIES AND SEQUENCES APPLICATIONS There are many situations in the real-world where sequences and series occur. When to use a sequence: If an amount steadily increases or decreases over time and follows a pattern, it could be represented by an arithmetic or geometric sequence. You know the sequence is an arithmetic sequence if the amounts increase by adding a number to get to the next one. You know the sequence is a geometric sequence if the amounts increase by adding a number to get to the next one. When to use a series: To use a series, the situation would require finding the sum of the amounts in a sequence (arithmetic or geometric). Arithmetic Sequence Example: EXAMPLES An artist is creating a triangular shaped sculpture made of metallic marbles. There are 44 of the marbles in the first row with two less in each row. How many marbles are in the 22 nd row? Solution: This situation is represented by an arithmetic sequence. Since the number of marbles decreases by 2, the difference is -2 and d = -2. The first term is 44, difference is -2, and n is 22.
Use the formula in your Information Booklet in Topic 2, Section 2.5 for finding the nth term in an arithmetic sequence. The number of marbles in the 22nd row is: 44 + 21(-2) The answer is: 2 marbles Arithmetic Series Example: Teddy is saving money in a savings account at a bank. His first deposit was 50 Euro. Each month after he opened the account, he deposits 10% more than the previous month. How much money does he have in the account after 8 months? Solution: Because he deposits 10% more each month, the quantities increase by adding 10% of 50 = 5 Euro each month. Since you are asked to find the total, you need to use the arithmetic series formula in your Information Booklet, Topic 2.5. Because we do not know the last term of the sequence, we would use the first of the two formulae with the first term is 50; the difference is 5, and n = 8. The amount of money in the account is: (4/2)(2(50) + 7(5)) The answer is: 270 Euros. Geometric Sequence Example: A ball is dropped from a height of 8 meters. With each bounce, the height decreases by 15%. What is the height of the ball after the third bounce?
Solution: This is represented by a geometric sequence. Since the height decreased by 15 %, the ratio, r = 85%. The first term is 8, ratio =.85, and n is 4 (not 3). Use the formula in your Information Booklet in Topic 2, Section 2.6 for finding the nth term in a geometric sequence. Draw a picture of the ball bouncing to verify the information you are using is correct. The height is: 8(.85^3) The answer is 4.91 m 3sf Geometric Series Example: A competitive runner increases his running distance by 9% each week. If he runs 5 kilometer the first week, find the total distance he ran after 8 weeks? Solution: This situation is represented by a geometric sequence because the quantities are increased by multiplying by 9%. You are asked to find the total distance. This means you are looking for the sum of the geometric sequence. To find the sum, you will use the formula for a geometric series. Use the formula for the sum of a geometric sequence in your Information Booklet in Section 2.6. Because the distance ran increases every day above the previous day s distance, the ratio will be 100% + 9%. Writing this is a decimal number is 1.09. The ratio is 1.09. Since the ratio is greater than 1, you will use the formula on the left in the Information Booklet with: first term = 5; r = 1.09; and n = 8
The sum is: 5(1.09^8 1) / (1.09 1) The answer is: 55.1 km 3sf Solving Sequences and Series Problems When the Number of Terms is Not Known 1. During 2009, the monthly sales of computers at Bob s Computers decreased at the rate of 11 per month. If they sold 245 in January, how many did they sell in October? 2. In which month did Bob Computers sell 102 computers? 3. Sally is selling t-shirts for her club. The set up cost is $20. The first t-shirt costs $8.00 and each additional t-shirt costs $.40 less. How many will 30 t- shirts cost? 4. The club treasurer told Sally that they have $95 in their account. How many t-shirts can they buy? 5. A ball was dropped at a height of 8 meters. It bounces to a height70% of each bounce. What is the height of the ball before the 5 th bounce? 6. How many times will the ball bounce before reach a height of 3 meters? ANSWERS
1. Arithmetic sequence with d = -11, the first term = 245, and n = 10. Answer is 146 computers. 2. Arithmetic sequence with d = -11, the first term = 245, and the last term is 100. Answer is February 2010 because n = 14. 3. Arithmetic Series with d = -.40, the first term = 8, and n = 30. Answer is $20 + $66 = $86. 4. Arithmetic Series with d = -.40, the first term = 8, and the last term is 75. Answer is 27 t-shirts. 5. Geometric Sequence with r =.7, the first term = 8, and n = 5. Answer is 1.92 m 3sf 6. Geometric Sequence with r =.7, the first term = 8, and the last term = 3. n = 3.75 which means 3 bounces