* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Slide 1 - Mrs. Hille`s FunZone
Survey
Document related concepts
Positional notation wikipedia , lookup
Ethnomathematics wikipedia , lookup
Infinitesimal wikipedia , lookup
Law of large numbers wikipedia , lookup
Georg Cantor's first set theory article wikipedia , lookup
History of logarithms wikipedia , lookup
Surreal number wikipedia , lookup
Proofs of Fermat's little theorem wikipedia , lookup
Large numbers wikipedia , lookup
Bernoulli number wikipedia , lookup
Location arithmetic wikipedia , lookup
Real number wikipedia , lookup
Elementary arithmetic wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Transcript
Look for a Pattern Example Since his first birthday, André’s uncle has given him a gift of money such that the number of dollars matched his age. Today is André’s fourteenth birthday. How much money has he been given over the years by his uncle? Example If André’s uncle continues giving him these gifts until Andre is 50, how much money will he have received by then? Think: Determine what you are trying to find. You want to know the total amount of money Andre received from this uncle in 14 years and how much he will have received after 50 years. Think: How could you find this sum? Is there an easier way than just adding the numbers from 1 to 14 and then adding the numbers from 1 to 50? Is there a pattern to this sum? Is there a rule based on the year number that can be used to find the sum in dollars? Look at the first few years to try to determine a pattern in the sums. Making a table may help you organize the information. # Sum in Dollars 1 2 3 4 5 6 7 1 1+2=3 1+2+3=6 1 + 2 + 3 + 4 = 10 1 + 2 + 3 + 4 + 5 = 15 1 + 2 + 3 + 4 + 5 + 6 = 21 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28 There definitely is a pattern in the sums. The next number in the sequence can be found in the previous number. Think: How can you find the sum of the numbers to 14 and 50 without writing down the entire list of numbers? Notice that the numbers that are added to get 28 in the table can be added in pairs that always give the same numerical result. That is, 1 + 7 = 8, 2 + 6 = 8, and 3 + 5 = 8. Since there are 7 numbers in the sum, there are 3 pairs of numbers and an unpaired middle number. The middle one is one half the sum of one of the pairs. Think: What will be the sum of the first and last terms in 1 + 2 + 3 + … + 13 + 14? How many pairs will there be in the sum? 1 + 14 = 15, as does 2 + 13. There are 7 pairs with no remainder. 14 years: 7 x 15 = $105 Another way to look at this same problem is to look at the sum of two sets of numbers from 1 to 7. The answer is obviously twice what is needed and requires you to divide by 2. sum 1 2 3 4 5 6 7 7 6 5 4 3 2 1 8 8 8 8 8 8 8 1+2+3+4+5+6+7 7(7 + 1) = 2 7x8 = 2 = 28 Likewise, considering the sum of two sets of numbers from 1 to 50, there is twice what is needed in the product of (1 + 50) and 50. Thus the sum 1 + 2 + 3 + 4 + 5 + … + 50 = 50(1 + 50) = 1,275 2 Think: How can this be generalized for a sum of the first n whole numbers? Following this same pattern, the sum of the first n whole numbers, 1 + 2 + 3 + 4 + 5 + … + n, can be found. Find the common sum (n + 1), multiply it by n, and, realizing that this is twice the needed sum, divide by 2 to get the correct answer. 1+2+3+4+5+…+n n(n + 1) = 2 Now try this out with some of your examples to convince yourself that it works. 1+2+3+4+5+6 6(6 + 1) = 2 = 21 This answer matches the value of the sum in the first six whole numbers in the table. Exercise Square numbers are numbers that can be displayed by a pattern of dots that form a square. Give the number of dots in each square shown. Exercise Draw the next square; then give the rule for finding the nth term of this sequence. Exercise Look at the pattern of sums given below. Write the next four patterns and their sums. 1=1 1+3=4 1+3+5=9 1 + 3 + 5 + 7 = 16 1 + 3 + 5 + 7 + 9 = 25 Write a rule for finding the sum based on the quantity of odd numbers n in the sum. Find the sum of the first 20 odd numbers by using your rule. Then use a calculator to check your sum by adding the first 20 odd numbers together. Exercise An eccentric math teacher told his class that he would assign one problem on the first day of school, two problems on the second day, four problems on the third day, and so on. At this rate, how many problems would he assign on the tenth day? Find a rule that can be used to determine the number of problems assigned on the nth day. Exercise Triangular numbers are numbers that can be displayed by a pattern of dots that form a triangle. The first four triangular numbers are 1, 3, 6, and 10. What are the next four triangular numbers? What is the rule for finding the value of a triangular number based on the number of the term in the sequence? Exercise Find the sum of the first eight positive multiples of 11. Write out this sum. If you add the first and last addends, then the second and next to last, and continue in this pattern, what is the common sum of each pair? How many pairs of addends are there in this sum? How does the number of pairs relate to the number of addends? What could you do with the common sum and the number of pairs to find the entire sum? Exercise Find the sum of the first ten natural numbers and then the first one hundred natural numbers. Give a rule that can be used to find the sum of the first n natural numbers. Exercise In the last two problems, the number of addends was always even. Verify that this same rule will work for finding the sum of the first five natural numbers. Do you think this rule works for all natural numbers? Exercise The following pattern is called the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, … Describe how to find the next term in the pattern using the terms that already exist. Find the next three terms of the sequence.