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Comprehensive Guide to Arithmetic and Geometric Sequences and Series

Explore arithmetic and geometric sequences, including common differences, ratios, nth terms, arithmetic and geometric means, series sums, and practical examples with exercises.

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Five Star bus Company (North Luzon Transit) is one of the many bus transportation companies in the Philippines servicing routes between Pampanga, and Metro Manila, Nueva Ecija or Pangasinan. One day, on its way back to its terminal at Lrt Monumento station, one (1) passenger went down at San Carlos, then, another four (4) passengers went down to Laug, seven (7) passengers went down to San Simon, Pampanga and ten (10) passengers went down to 7th avenue
QUESTIONS •List down the number of passengers who went down in each place. 1, 4, 7, 10 •Does it form sequence? If it does, how is the sequence formed?
ARITHMETIC SEQUENCE
ARITHMETIC SEQUENCE • An arithmetic sequence is a sequence whose consecutive terms have a common difference. • Common difference (d) is the constant number added to the preceding term of the arithmetic sequence. It can be calculated by subtracting any two consecutive terms in the arithmetic sequence. • Each element or object in the sequence is called term.
ARITHMETIC SEQUENCE Common difference(d) - The constant number added to the preceding term.  We can also get this by subtracting the first term from the second term. 3 3 3
EXAMPLE What is the common difference of each sequence? 1.) 5 25 45 65 85 2.) 4 8 16 32 64 3.) 3 17 31 45 59 4.) 17 15 13 11 9 d = 20 d = -2 d = 14 N. A .S
EXAMPLE Get the next three terms by adding the common difference. 1.) 5 25 45 65 85 2.) 3 17 31 45 59 3.) 17 15 13 11 9 105 125 145 73 87 101 7 5 3
nth Term of an Arithmetic Sequence
N t h Te r m o f A r i t h m e t i c S e q u e n c e The formula for the nth term of an arithmetic sequence is , Where: = first term = last term = number of terms d = common difference
E X A M P L E 1 • Find the 15th term of the arithmetic sequence: 12, 16, 20, 24, 28 𝑎𝑛=𝑎1+(𝑛− 1) 𝑑
Find the 20th term of the sequence whose first term is 6 and whose seventh term is 42. 𝑎𝑛=𝑎1+(𝑛− 1) 𝑑 EXAMPLE 2
Find the 20th term of the sequence whose first term is 6 and whose seventh term is 42. 𝑎𝑛=𝑎1+(𝑛− 1) 𝑑 EXAMPLE 2
𝑎1=6 𝑑=6 𝑎𝑛=𝑎1+(𝑛− 1) 𝑑 Find the 20th term of the sequence whose first term is 6 and whose seventh term is 42.
EXAMPLE 3 In the arithmetic sequence 11, 19, 27, 35, 43, … which term is 147?
11, 19, 27, 35, 43,….. 147 𝑎𝑛=𝑎1+(𝑛− 1) 𝑑
11, 19, 27, 35, 43,….. 147 𝑎𝑛=𝑎1+(𝑛− 1) 𝑑
ARITHMETIC MEANS
Arithmetic Means The terms between two nonconsecutive terms
Find the missing terms in each of the following arithmetic sequence. 9, 17, ___, ___, ___, 49 7, 9, ___, ___, 15 25 33 41 11 13 ARITHMETIC MEANS
Insert three arithmetic means between 18 and 30. Step 1. Identify the given using the arithmetic sequence formula. 𝒂𝒏 =𝒂𝟏+(𝒏−𝟏) 𝒅 𝒂𝒏=𝟑𝟎
Insert three arithmetic means between 18 and 30. Step 2. Find the common difference.
Insert three arithmetic means between 18 and 30. Since d=3, 18, 21, 24, 27, 30
Find the four arithmetic means in the sequence: a+2b, 3b, 4b-a, ___, ___, ___, ____, 9b-6a. Find the common difference. Note: Subtract the first term from the second term.
a+2b, 3b, 4b-a, ___, ___, ___, ____, 9b-6a. Since d=b-a, a+2b, 3b, 4b-a, 5b-2a, 6b- 3a, ___, ____, 9b-6a.
a+2b, 3b, 4b-a, ___, ___, ___, ____, 9b-6a. Since d=b-a, a+2b, 3b, 4b-a, 5b-2a, 6b- 3a, ______,______, 9b-6a. 7b-4a 8b-5a
ARITHMETIC MEAN
Arithmetic Mean It is the average or the sum of a collection of numbers divided by the number of numbers in collection
𝑨𝒓𝒊𝒕𝒉𝒎𝒆𝒕𝒊𝒄 𝑴𝒆𝒂𝒏 𝑨𝒓𝒊𝒕𝒉𝒎𝒆𝒕𝒊𝒄 𝒎𝒆𝒂𝒏= 𝒔𝒖𝒎𝒐𝒇 𝒂𝒄𝒐𝒍𝒍𝒆𝒄𝒕𝒊𝒐𝒏𝒐𝒇 𝒏𝒖𝒎𝒃𝒆𝒓𝒔 𝒏𝒐 .𝒐𝒇 𝒏𝒖𝒎𝒃𝒆𝒓𝒔 𝒊𝒏𝒄𝒐𝒍𝒍𝒆𝒄𝒕𝒊𝒐𝒏 𝒙= 𝒙+ 𝒚 𝟐 FORMULA:
EXAMPLE Find the arithmetic mean between 8 and 20. 𝒙= 𝒙+ 𝒚 𝟐 The arithmetic mean between 8 and 20 is 14.
Find the arithmetic mean between and
Dominic’s score in five math test are 94, 90, 96, 93 and 95. What is his average score? 𝒙 = ∑ 𝒙 𝒏
SEATWORK
SEATWORK Determine the indicated term in the following arithmetic sequences. 1. 2. 3. 4. 5.
SEATWORK Determine the indicated term in the following arithmetic sequences. 1. 2. 3. 4. 5.
SEATWORK Determine the indicated term in the following arithmetic sequences. 1. 2. 3. 4. 5.
SEATWORK Determine the indicated term in the following arithmetic sequences. 1. 2. 3. 4. 5.
SEATWORK # 6-8, determine the arithmetic mean between the given numbers. # 9 and 10, determine the arithmetic means (AM) as indicated. 6. 8 and 30 9. Three AM between 12 and 44. 7. 16 and 35 10. Five AM between -2 and 16. 8. and
SEATWORK # 6-8, determine the arithmetic mean between the given numbers. # 9 and 10, determine the arithmetic means (AM) as indicated. 6. 8 and 30 9. Three AM between 12 and 44. 7. 16 and 35 10. Five AM between -2 and 16. 8. and
ARITHMETIC SERIES
Find the sum of the terms of a given sequence Lesson Objectives Determine Arithmetic Series
ARITHMETIC SERIES
ARITHMETIC SERIES Is the sum of the terms of an arithmetic sequence. The sum of the terms of an arithmetic sequence. , can be found as: or
ARITHMETIC SERIES Where:
ARITHMETIC SERIES Where:
SIGMA NOTATION A more concise way to express the sum of is to use the summation notation or sigma notation. The Greek letter Σ (sigma) tells us to sum or add up the terms. ∑ 𝒌 𝒏 (𝟑 𝒌) end (upper limit) start (lower limit) index of summation
EXAMPLE 1: Determine the sum of the first 10 terms of the arithmetic sequence 4, 15, 26, 37, … or
EXAMPLE 2: There is a stack of boxes in a warehouse. There are 20 boxes in the bottom layer, 19 in the second, 18 in the third, and so on. The last layer has only one box. How many boxes are there in the stack? or
EXAMPLE 3: Find the sum of the first 20 terms of the arithmetic sequence 11, 16, 21, 26, 31, … Steps: • Find the common difference • Find the last term • Find the sum of the arithmetic sequence
EXAMPLE 3: Find the sum of the first 20 terms of the arithmetic sequence 11, 16, 21, 26, 31, … • Find the common difference • Find the last term
EXAMPLE 3: Find the sum of the first 20 terms of the arithmetic sequence 11, 16, 21, 26, 31, … Find with
EXAMPLE 3: Find the sum of the first 20 terms of the arithmetic sequence 11, 16, 21, 26, 31, … Given:
Find the sum of the first 20 terms of the arithmetic sequence 11, 16, 21, 26, 31, … 𝑛=20 𝑎1=11 𝑎𝑛 =106
EXAMPLE 4: Find the sum of the first 38 terms of the arithmetic sequence if the first term is 18 and the 5th term is 38
EXAMPLE 5: ∑ 𝒌=𝟑 𝟖 (𝟏𝟓−𝟐𝒌)
EXAMPLE 6: ∑ 𝒌=𝟏 𝒏 (𝟐𝒌+𝟓)=𝟓𝟓 ,𝑭𝒊𝒏𝒅 𝒕𝒉𝒆 𝒗𝒂𝒍𝒖𝒆𝒐𝒇 𝒏
Do you have any questions?
Seatwork: 1. 2. 3. 4. 5.
Seatwork:
Thank you for listening!
GEOMETRIC SEQUENCE
◦A geometric sequence is a sequence in which each term is obtained by multiplying the preceding term by fixed number. ◦A sequence is geometric if there exist a number r, called the common ratio. ◦Common ratio, r can be determined by dividing any term in the sequence by the term that precedes it.
Common ratio, r can be determined by dividing any term in the sequence by the term that precedes it.
LET’S TRY THESE EXAMPLES! Identify the common ratio of the following: 1.) 1, 2, 4, 8…
2.) 80 , 20, 5, ... 3.) 2, -8, 32, -128, …
4.) 3, 24, 192, 1 536, … 5.) 9, 9, 9, 9, … 6.) 1, 8, 64, 512, …
7.) 160, 80, 40, 20, 10...
Tell whether or not each of the following sequence is an arithmetic or a geometric sequence. Give the common ratio for sequences that are geometric and the common difference for arithmetic sequence. a)7, 14, 21, 28, 35 ... b)3, 15, 75, 375, 1 875... c)1, 16, 31, 46, 61 ... d)160, 80, 40, 20, 10, 5, 2.5
ANSWERS: a) 7, 14, 21, 28, 35 ... ARITHMETIC; COMMON DIFFERENCE = 7, b) 3, 15, 75, 375, 1 875... GEOMETRIC; COMMON RATIO = 5 c) 1, 16, 31, 46, 61 ... ARITHMETIC; COMMON DIFFERENCE = 15, d) 160, 80, 40, 20, 10, 5, 2.5 GEOMETRIC; COMMON RATIO
Nth term of Geometric Sequences
The nth term of a geometric sequence •The nth term of a geometric sequence is given by: Where: .
Examples: •What is the 10th term of the geometric sequence 8, 4, 2, 1? 1st step: find the common ratio: 𝑟 = 𝑎2 𝑎1 𝑟 = 4 8 𝑜 𝑟 𝟏 𝟐
Examples: •What is the 10th term of the geometric sequence 8, 4, 2, 1? Find the nth term: 𝑟 = 1 2 𝑎10=8( 1 512 ) 𝒂𝟏𝟎= 𝟏 𝟔𝟒
Examples: •Find the missing term in 3, 12, 48,__, __ Find r:
Examples: •Find the missing terms in __, __ 32, 64, 128. Find r:
Examples: •Find the missing terms in __, __ 32, 64, 128. r = 2;
Find the 1st term of a geometric sequence whose 4th term is and whose 7th term is .
GEOMETRIC MEANS
Geometric means is/are the missing term(s) between two nonconsecutive terms in a geometric sequence
Example 1. What is the two geometric means between 1 and 27 in the sequence; 1, 3, 9, 27, 81, 243 ... ? Answer: 3 and 9
Example 2. What is the three geometric means between 2 and 32 in the sequence; 2, 4, 8, 16, 32... ? Answer: 4, 8, 16
Geometric mean between any two positive numbers a and b is the square root of their product.
Example 4. Find the positive geometric mean between 4 and 25.
Example 5. Find the positive geometric mean between -3 and -12.
Example 6. Insert three geometric means between 5 and 1280. 5, __, __, __, 1 280 5, __, 80, __, 1 280
Example 6. Insert three geometric means between 5 and 1280. 5, __, 80, __, 1 280 5, 20, 80, __, 1 280
Example 6. Insert three geometric means between 5 and 1280. 5, 20, 80, __, 1 280 5, 20, 80, 320, 1 280
QUESTION Insert three geometric means in the sequence , __, __, __, 4
QUESTION Insert two geometric means in the sequence 1, __, __, 216
QUESTION Insert two geometric means in the sequence 20, __, __, 1 280 Steps: • Find the common ratio:
QUESTION Insert two geometric means in the sequence 20, __, __, 1 280
QUESTION Insert two geometric means in the sequence 20, __, __, 1 280 • Complete the sequence by multiplying the common ratio. r=4 20, 80, 320, 1 280
QUESTION Insert four geometric means in the sequence 2.5, __, __, __, __, -2560
TEN RAINBOW FACTS OF Finite and Infinite GEOMETRIC SERIES