lesson1-math10-w1q1arithmeticsequencesandseries-220919084054-a2d23a2a.pptx

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1. Arithmetic Sequences and Series

2. Arithmetic Sequence Arithmetic sequence is a set of numbers listed in a specific order such that the difference between succeeding terms is constant is a sequence in which each term is obtained from the preceding term by adding a common difference. Term Each number in the list of the sequence and is denoted by , …, , where is the nth term in the sequence.  

3. Common difference This is any two consecutive terms have a constant difference of the given arithmetic sequence and denoted by d nth Term of an Arithmetic Sequence n 1 2 3 4 5 f (n) 30 25 20 15 10 + (n-1)d   the n th term the first term a multiple that is less than n the common difference

4. Use the formula for Arithmetic Sequence = + ( n - 1 ) d   Example 1: Find the 47 th term given the sequence of numbers 5, 13 , 21 , 29, 37 Solution: Given: = ; = ; n = 47 ; d = 8   = + ( n - 1 ) d   = + ( 47 - 1 ) 8   = + ( 46) 8   = + 368   =  

5. Use the formula d = -   Example 2: Find the 12 th term in the sequence 11x-5 , 14x-2 , 17x +1 , :Solution Given: = ; = ; n = 12 ; d = (14x -2) – (11x – 5)   Solve for the common difference d = (14x -2) – (11x – 5) d = 14x -2– 11x + 5 d = 14x -2– 11x + 5 d = 14x – 11x -2 + 5 d = 3x + 3 Solve for the 12 th term = + ( n - 1 ) d   = 11x - 5 + ( 12 - 1 ) (3x + 3)   = 11x - 5 + ( 11 ) (3x + 3)   = 11x - 5 + (33x + 33)   = 11x - 5 + 33x + 33   = 44x + 28  

6. = + ( n - 1 ) d   Example 3: In the arithmetic sequence 21, 16, 11, 6, …, which term is -34 Solution: Given: = -34 ; = 21 ; n = ? ; d = -5   Solve for number of terms: = + ( n - 1 ) d   = + ( n - 1 ) -5   = -5 n + 5   -21-5= -5 n   = -5 n   =     Thus, -34 is the 12 th term:

7. Arithmetic Means Also known as average, is a number calculated by adding two numbers a and b and dividing by the number of terms in the set denoted by   = = 21   Insert an arithmetic mean between 15 and 27 Solution 2: Solve for d = + ( n - 1 ) d   27 = + ( 3 - 1 ) d   27 -15= 2 d d = 6 Adding the common difference, 6, with the first term, 15 is the arithmetic mean. Thus, the arithmetic mean is 15+6 = 21

8. Arithmetic Series The sum of the terms in an arithmetic sequence given by: = ( + )   Where is the sum of the terms;   is the first term;   is the common difference;   n is the number of terms.  

9. = + ( n - 1 ) d   Example 4: Find the sum of the first 24 terms of an arithmetic sequence where the first term is 7 and the common difference is 5. Find   = + ( 24 - 1 ) 5   = + ( 23 ) 5   = + 115   = 122   Then , find the sum. = ( + )   = ( + )   = ( 129 )   = ( 129 )   =  

10. Practic e Exercises: 1. Given the sequence of numbers 15 , 9 , 3 , -3 , -9 , …., find the 73 rd term. 2. Find the 28 th term given that = -1 and d = 5   3. The seat plan of a concert venue is arranged in a sequence. The first row has 5 sets. The seats for the succeeding row are 3 more than the previous. If there are 42 rows, what is the seating capacity of the concert venue? Find the um of the arithmetic sequence