Find the next sequence of entries in the 11th row of the Pascal’s triangle. An ice cream parlor has 12 flavors of ice cream and with toppings chocolate, nuts, chips, raisins, M&M and crushed. How many different sets of a. Exactly 3 toppings could you have? b. Exactly 4 toppings c. Any number of toppings ( up to 6 available)? Arithmetic Sequences and Series Sequences List with commas 3, 8, 13, 18 Series “Indicated sum” 3 + 8 + 13 + 18 An Arithmetic Sequence is defined as a sequence in which there is a common difference between consecutive terms. Which of the following sequences are arithmetic? Identify the common difference. 3, 1, 1, 3, 5, 7, 9, . . . YES 15.5, 14, 12.5, 11, 9.5, 8, . . . d 2 YES d 1.5 84, 80, 74, 66, 56, 44, . . . NO 8, 6, 4, 2, 0, . . . NO 50, 44, 38, 32, 26, . . . YES d 6 26, 21, 16, 11, 6, . . . The general form of an ARITHMETIC sequence. First Term: a1 Second Term: a2 a1 d Third Term: a3 a1 2d Fourth Term: a4 a1 3d Fifth Term: a5 a1 4d nth Term: an a1 n 1 d Formula for the nth term of an ARITHMETIC sequence. an a1 n 1 d an The nth term a1 The 1st term n The term number d The common difference Given: 79, 75, 71, 67, 63, . . . Find: a32 IDENTIFY a1 79 SOLVE an a1 n 1 d d 4 a32 79 32 1 4 n 32 a32 45 Given: 79, 75, 71, 67, 63, . . . Find: What term number is -169? IDENTIFY a1 79 d 4 an 169 SOLVE an a1 n 1 d 169 79 n 1 4 n 63 Given: a10 3.25 Find: a1 a12 4.25 Why?!!? What’s the real question? Letting 3.25 and 4.25 in new position to find the difference a1 3.25 a3 4.25 n3 The Difference SOLVE an a1 n 1 d 4.25 3.25 3 1 d d 0.5 The common difference is always the difference between any term and the term that proceeds that term. Given: a10 3.25 Find: a1 a12 4.25 IDENTIFY Use the original position of the term to find the first term SOLVE an a1 n 1 d a10 3.25 d 0.5 3.25 a1 10 1 0.5 n 10 a1 1.25 You can use either the two given 3.25 or 4.25 to find the first term. -1.25 -0.75 -0.25 0.25 0.75 1.25 1.75 2.25 2.75 3.25 3.75 4.25 When a garbage truck starts collecting rubbish it ﬁrst stops at a corner store where it collects 86 kg of rubbish. It then travels down a long suburban street where it picks up 40 kg of rubbish at each house. a. How much garbage would be carried by the truck after: i 15 pick-ups? ii 27 pick-ups? b. The maximum amount of garbage that can be carried by the truck is 1500 kg. After picking up from the corner store, what is the maximum number of houses it can pick up from before it is fully loaded? Answers: i. 50 73 2 p 71 69 67 . . . 25 27 p 1 71 69 67 . . . 25 27 27 25 . . . 67 69 71 44 44 44 . . . 44 44 44 50 Terms 50 71 27 2 1100 71 + (-27) Each sum is the same. a1 a1 d a1 2d . . . a1 n 1 d a n 1 d . . . a 1 1 2d a1 d a1 a a n 1 d a a n 1 d . . . a a n 1 d 1 s 1 1 n a1 an 2 1 1 1 S Sum n Number of Terms a1 First Term an Last Term S Find the sum of the terms of this arithmetic series. 35 29 3k n a1 an k 1 2 n 35 a1 26 a35 76 35 26 76 S 2 S 875 Find the sum of the terms of this arithmetic series. 151 147 143 139 . . . 5 n a1 an S 2 n 40 a1 151 a40 5 What term is -5? an a1 n 1 d 5 151 n 1 4 n 40 40 151 5 S 2 S 2920 Substitute an a1 n 1 d n a1 an S 2 S S n a1 a1 n 1 d 2 n 2a1 n 1 d 2 n # of Terms a1 1st Term d Difference 36 Find the sum of this series 2.25 0.75 j j 0 2.25 3 3.73 4.5 . . . S n 2a1 n 1 d n 37 2 37 2 2.25 37 1 0.75 a1 2.25 S d 0.75 S 582.75 2 35 45 5i n a1 an S 2 n 35 a1 40 an 130 35 40 130 S 2 S 1575 i 1 S n 2a1 n 1 d 2 n 35 a1 40 d 5 S 35 2 40 35 1 3 S 1575 2 An introduction………… 1, 4, 7, 10, 13 35 2, 4, 8, 16, 32 62 9, 1, 7, 15 12 9, 3, 1, 1/ 3 20 / 3 6.2, 6.6, 7, 7.4 27.2 , 3, 6 3 9 1, 1/ 4, 1/16, 1/ 64 85 / 64 9.75 , 2.5, 6.25 Arithmetic Sequences Geometric Sequences ADD To get next term MULTIPLY To get next term Arithmetic Series Sum of Terms Geometric Series Sum of Terms Find the next four terms of –9, -2, 5, … Arithmetic Sequence 2 9 5 2 7 7 is referred to as the common difference (d) Common Difference (d) – what we ADD to get next term Next four terms……12, 19, 26, 33 Find the next four terms of 0, 7, 14, … Arithmetic Sequence, d = 7 21, 28, 35, 42 Find the next four terms of x, 2x, 3x, … Arithmetic Sequence, d = x 4x, 5x, 6x, 7x Find the next four terms of 5k, -k, -7k, … Arithmetic Sequence, d = -6k -13k, -19k, -25k, -32k Vocabulary of Sequences (Universal) a1 First term an nth term n number of terms Sn sum of n terms d common difference nth term of arithmetic sequence an a1 n 1 d sum of n terms of arithmetic sequence Sn n a1 an 2 Given an arithmetic sequence with a15 38 and d 3, find a1. x a1 First term 38 an nth term 15 n number of terms NA Sn sum of n terms -3 d common difference an a1 n 1 d 38 x 15 1 3 X = 80 Find S63 of 19, 13, 7,... -19 a1 First term 353 ?? an nth term n number of terms 63 x Sn sum of n terms 6 d common difference an a1 n 1 d ?? 19 63 1 6 ?? 353 n a1 an 2 63 19 353 2 Sn S63 S63 10521 Try this one: Find a16 if a1 1.5 and d 0.5 1.5 a1 First term x 16 an nth term n number of terms NA Sn sum of n terms 0.5 d common difference an a1 n 1 d a16 1.5 16 1 0.5 a16 9 Find n if an 633, a1 9, and d 24 9 a1 First term 633 an nth term x n number of terms NA Sn sum of n terms 24 d common difference an a1 n 1 d 633 9 x 1 24 633 9 24x 24 X = 27 Find d if a1 6 and a29 20 -6 a1 First term 20 an nth term 29 n number of terms NA Sn sum of n terms x d common difference an a1 n 1 d 20 6 29 1 x 26 28x 13 x 14 Find two arithmetic means between –4 and 5 -4, ____, ____, 5 -4 a1 First term 5 an nth term n number of terms 4 NA x Sn sum of n terms d common difference an a1 n 1 d 5 4 4 1 x x 3 The two arithmetic means are –1 and 2, since –4, -1, 2, 5 forms an arithmetic sequence Find three arithmetic means between 1 and 4 1, ____, ____, ____, 4 1 a1 First term 4 an nth term 5 NA x n number of terms Sn sum of n terms d common difference an a1 n 1 d 4 1 5 1 x 3 x 4 The three arithmetic means are 7/4, 10/4, and 13/4 since 1, 7/4, 10/4, 13/4, 4 forms an arithmetic sequence Find n for the series in which a1 5, d 3, Sn 440 5 a1 First term y an nth term x n number of terms 440 Sn sum of n terms 3 d common difference an a1 n 1 d y 5 x 1 3 x 440 5 5 x 1 3 2 x 7 3x 440 2 880 x 7 3x 0 3x 2 7x 880 Graph on positive window X = 16 n Sn a1 an 2 x 440 5 y 2 The sum of the first n terms of an infinite sequence is called the nth partial sum. Sn n (a1 an) 2 Example 6. Find the 150th partial sum of the arithmetic sequence, 5, 16, 27, 38, 49, … a1 5 d 11 c 5 11 6 an 11n 6 a150 11150 6 1644 S150 150 5 1644 75 1649 123,675 2 Example 7. An auditorium has 20 rows of seats. There are 20 seats in the first row, 21 seats in the second row, 22 seats in the third row, and so on. How many seats are there in all 20 rows? d 1 c 20 1 19 an a1 n 1 d a20 20 19 1 39 20 S 20 20 39 10 59 590 2 Example 8. A small business sells $10,000 worth of sports memorabilia during its first year. The owner of the business has set a goal of increasing annual sales by $7500 each year for 19 years. Assuming that the goal is met, find the total sales during the first 20 years this business is in operation. a1 10,000 d 7500 c 10,000 7500 2500 an a1 n 1 d a20 10,000 19 7500 152,500 20 S20 10,000 152,500 10 162,500 1,625,000 2 So the total sales for the first 2o years is $1,625,000 Geometric progressions We shall now move on to the other type of sequence we want to explore. Consider the sequence 2, 6, 18, 54, ... . Here, each term in the sequence is 3 times the previous term. And in the sequence 1, −2, 4, −8, ... , each term is ___ times the previous term. Sequences such as these are called geometric progressions, or GPs for short. Exercises (a) Write down the ﬁrst ﬁve terms of the geometric progression which has ﬁrst term 1 and common ratio 1/2. (a) Find the 10th and 20th terms of the GP with ﬁrst term 3 and common ratio 2. The sum of a geometric series Find the sum of the geometric series 2 + 6 + 18 + 54 + ... where there are 16 terms in the series. For this series, we have a = 2, r = 3 and n = 6. Answer Find the sum of the geometric series 8, −4, 2, −1 + ... where there are 5 terms in the series. Solution For this series, we have a = 8, r = and n = 5. Answer 5 ½ How many terms are there in the geometric progression 2, 4, 8, ..., 128? Solution In this sequence a = 2 and r = 2. We also know that the n-th term is 128. Answer n = 7 Example How many terms are there in the geometric progression 2, 4, 8, ..., 128? Answer n = 7. Use the rule to determine the 8th and 11th term of the geometric sequence 100,50,25,12.5,... The 8th term is ---- A fisherman harvested 350 kg of fishes on Monday. From Monday to Friday, the amount of fishes, he harvested increased by 10% per day. What’s the total amount of fishes did the fisherman harvest in the first five days? Round your answer to the nearest whole number. A population of a bacteria doubles its numbers every minute. If we start off with ﬁve bacteria, how many will we have at the start of the 31st minute (that is, after half an hour)? Find the sum of the arithmetic series. 1)-4 - 1 + 2 + 5 + 8 + 11 + 14 2)13 + 15 + 17 + 19 + ... + 31 3) 247 + 245 + 243 + 241 + ... + 229 4) -16 - 11 - 6 - 1 + ... + 39 Find the sum of the geometric series. 1. 4/3 + 16/3 + 64 /3 + 256/ 3 + 1024 /3 2. 3 + 9 + 27 + 81 + 243 3. 3/ 2 + 3/ 8 + 3 /32 + 3 /128 + 3 /512 The sum of the numbers in arithmetic progression is 24. If the first number is decreased by 1 and the second decreased by 2, the three numbers are in a geometric progression. Find the three numbers. Find the sum of the geometric series. 1. 4/3 + 16/3 + 64 /3 + 256/ 3 + 1024 /3 2. 3 + 9 + 27 + 81 + 243 3. 3/ 2 + 3/ 8 + 3 /32 + 3 /128 + 3 /512 4. How many terms are there in the geometric progression 2.3, 10.35, 46.575,+ ... + 158,590,804.2 ? 5. A population of a bacteria doubles its numbers every minute. If we start off with ﬁve bacteria, how many will we have at the start of the 31st minute (that is, after half an hour)? 7. In a geometric sequence, the 4th term is 24 and the 9th term is 768. Find three terms of the sequence.