4/1/2017. PS. Sequences and Series FROM 9.2 AND 9.3 IN THE BOOK AS WELL AS FROM OTHER SOURCES. TODAY IS NATIONAL MANATEE APPRECIATION DAY

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1 PS. Sequences and Series FROM 9.2 AND 9.3 IN THE BOOK AS WELL AS FROM OTHER SOURCES. TODAY IS NATIONAL MANATEE APPRECIATION DAY 1

2 Oh the things you should learn How to recognize and write arithmetic sequences How to find an nth partial sum of an arithmetic sequence How to use arithmetic sequences to model and solve real-life problems How to recognize, write, and use geometric sequences. Understand series and their sums. ARITHMETIC SEQUENCES An arithmetic sequence is a sequence in which each term differs from the previous one by the same fixed number. The common difference of an arithmetic sequence is... The difference d between consecutive terms of an arithmetic sequence Note: It can also be referred to as an arithmetic progression. IB BOOK is arithmetic, (note the double arrow is IFF) for all positive integers n where d is a constant called the common difference. PRECALCULUS BOOK, where d is the common difference between consecutive terms of the sequence, and. Therefore, an arithmetic sequence may be thought of as linear function whose domain is the set of natural numbers. 2

3 Example: see if the difference is constant! Determine whether or not the following sequence is arithmetic. If it is, find the common difference. 7, 3, - 1, - 5, - 9,... Find a formula for the nth term of the arithmetic sequence whose common difference is 2 and whose first term is C=

4 Example New one Find a formula for the nth term of the arithmetic sequence whose common difference is 4 and whose fifth term is 19. The General Term formula IB BOOK Suppose the first term of an arithmetic sequence is and the common difference is d. Then, 2, 3, and so on. Hence SO: In General, For an arithmetic sequence with first term and common difference d the general term or nth term is Precalculus Book: The nth term of an arithmetic sequence has the alternative recursive formula 4

5 Examples! 1) Find the sixth term of the arithmetic sequence that begins with 15 and ) Find the nth term of the arithmetic sequence with fifth term 19 and ninth term Another way to look at it. 4, 4! So , then d 2. This means we can find, because 4 You try it out! IB terms a)find the general term for an arithmetic sequence with 8 and

6 The Sum of a Finite Arithmetic Sequence We denote the sum of a arithmetic sequence as in both books! Thank goodness!!! However, the formulas are slightly different. This is how it s found So The sum of an arithmetic series with n terms is Another form: Example Find the 15 th partial sum of the sequence 2, 5, 8, 11, Find the sum of the first 20 terms of the sequence with nth term 28 5 D= -5, c=

7 Applications Each row of an auditorium has two more seats than the preceding row. Find the seating capacity of the auditorium if the front row has 30 seats and there are 40 rows. Homework: p.635 & odd, 33-43odd, 59-63odd, 67,69 9.3: Geometric Sequence A sequence is geometric if each term can be obtained from the previous one by multiplying by the same non-zero constant. A geometric sequence can also be referred to as a geometric progression. is geometric, constant called the common ratio. for all positive integers n where r is a Example: find the common ratio for the geometric sequence: 2,-10, 50, -250, 7

8 The General Term Formula For a geometric sequence with first term or and common ratio r, the general term or nth term is or Example: Find the nth term for the geometric sequence with first term 5 and common ratio 2. Example: Find the twentieth term of the geometric sequence 1, 3, 9, 27, 1 3 1,162,261,467 Harder examples: what we can do! Find the fifteenth term of the geometric sequence with a third term of and a sixth term of Another (better) way: , hence. Then use to find 8

9 Now you try A geometric sequence has 6 and 162. Find its general term. There is one other way So 27 3 whatever way works best. Finish out the problem. On Your Own A geometric sequence has 24 and 192. Find its general term. 9

10 Finite & Infinite sum A geometric series Is the addition of successive terms of a geometric sequence. The Sum of a Finite Geometric Sequence: A geometric series Is the addition of successive terms of a geometric sequence. If r < 1, the sum of the infinite geometric series: Examples Find the sum of to 12 terms. Identify the terms that are important for a finite sum. 10

11 Another example Find a formula for for the first n terms of Another example Find a) a formula for for the first n terms of B) to 12 terms 11