UNIT 3 VOCABULARY: SEQUENCES

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1 3º ESO Bilingüe Página UNIT 3 VOCABULARY: SEQUENCES.. Sequences of real numbers A sequence of real numbers is a set of real numbers that are in order. For example: 3, 5, 7, 9,, 3... is a set of numbers that are in order. They form a sequence: Order st nd 3rd 4th 5th 6th... Number A sequence usually has a rule, which is a way to find the value of each number... Terms of a sequence The terms of a sequence are each one of the elements that form the sequence. To make it easier to identify each term of a sequence, we often use this special style: For example, we can represent the sequence, -4, 6, -8, 0... as a, a, a 3, a 4, a 5... a = It means that number is the first term of the sequence. a = 4 It means that number -4 is the second term of the sequence. a 3 =6 It means that number 6 is the third term of the sequence. a 4 = 8 It means that number -8 is the fourth term of the sequence....

2 3º ESO Bilingüe Página.3. General term of a sequence The general term of a sequence is a formula that expresses the relationship between the value of every term and its position in the sequence. For example, a formula for the sequence 3, 5, 7, 9,... can be written like this: Look: =n+ a = +=3 a = +=5 a 3 = 3+=7 a 4 = 4+=9 You can calculate any term just by substituting "n" for the position you want to find out!!! a 6 = 6+=53 a 00 = 00+=0 a 5 = 5+=45. Find three more terms of these sequences: a) 3, 0, 7,... b), 6, 8,... c) -, 7, -49,... d) 40, 0, 0,.... Write the first five terms of the sequence related to every situation: a) Every month, you want yo go to the cinema twice as much as the month before. This month you've only been three times!!! b) You want to study for two more hours every week, because last time you only studied for one hour and you failed all of your exams. c) You have been growing three centimeters every year since you were.4m tall. 3. For every sequence write the relationship between each term and the previous one: a) 5, 8,, 4... c) 4,, 36, b) -0, 0, -0, 0... d) 9, 5,, -3, Fin the general term of the following sequences: a) 6,, 8, 4, c) 4, 5, 6, 7, 8... e), 4, 9, 6, 5, b), -,, -,, -... d) 4, 7, 0, 3, 6... f), 4, 8, 6, Find the first three terms of these sequences: a) =3n b) b n = n+4 c) c n = 5n d) d n = 3n e) e n =( 0) n Triangular Number sequence Fibonacci sequence

3 3º ESO Bilingüe Página 3.. Arithmetic sequences An arithmetic sequence or an arithmetic progression is a sequence in which the difference between one term and the next is always the same. In other words, to find the terms of the sequence you just add (or subtract) the same value each time For example:, 4, 7, 0, 3, 6, 9,, 5... is an arithmetic progression. This sequence has a difference of 3 between each number. 5, 3,, -, -3, is another arithmetic sequence. This sequence has a difference of - between each number. The difference between two terms in an arithmetic sequence is called the "common difference". It is often represented by d.. Select the arithmetic progressions and calculate the corresponding common difference:. Determine whether the following sequences are arithmetic and indicate the common difference: a) 5, -, -7. 3,... b), 3, 7, 8,, c),,, 4,... d), 3,, 5,3, 7, Write three more terms for these sequences: a) a =5, d = 9 b) b = 3, d = - c) c =0, d = 0.3 d) d =8, d = -5.. General term of an arithmetic sequence The general term of an arithmetic progression is: =a +d (n ) For example, for the sequence 3, 8, 3, 8, 3, 8, 33, 38,... The values of a and d are: a =3 (the first term) d = 5 (the "common difference")

4 3º ESO Bilingüe Página 4 The general rule can be calculated: =a +d (n )=3+5 (n )=3+5n 5=5n =5n. Find out the hundredth term of these arithmetic progressions: a) a =5, d = 9 b) b = 3, d = - c) c =0, d = 0.3 d) d =8, d = -5. Find the 0th of the progression: -3,, 5, 9,.3. Sum of terms The sum of the first n terms of an arithmetic sequence, S n, can be calculated: S n = a + n For example, let's add up the first 0 terms of the arithmetic sequence, 4, 7, 0, 3,... The first term a is, the commond difference d is 3 and we want to add up 0 terms (n = 0). So: S 0 = a +a 0 0= +a 0 0 a 0 =+3 (0 )=+3 9=8 S 0 = +8 0=9 0 =45 Why don't you add up the terms yourself, and see if it comes to 45?. Find the sum of the first 00 terms of the progression:, 6,, 6,. Find the sum of the first 50 terms of the progression 5, 3,, -, -3, -5, Find the sum of the first 000 natural numbers.

5 3º ESO Bilingüe Página Geometric Sequences An geometric sequence or a geometric progression is a sequence in which each term is found by multiplying the previous term by a constant. In other words, to find the terms of the sequence you just multiply (or divide) by the same value each time. For example:, 4, 8, 6, 3, 64,... is a geometric progression. Each term (except the first term) is found by multiplying the previous term by. 9, 3,, 3,,... is another geometric sequence. Each term (except the first 9 7 term) is found by dividing the previous term by 3. The factor between two terms in a geometric sequence is called the "common ratio". It is often represented by r. In the first example, the common ratio is r = In the second example, the common ratio is r = 3. Match each geometric progression with the corresponding fifth term:, 0, , 75, ,, -36, ,, Determine whether the following sequences are geometric and indicate the common ratio: a), -4, 8. 6,... b), 3, 7, 8,, c),,, 4,... d), 3, 9 4, 7 8, Write three more terms for these sequences: a) a =5, r = 5 b) b = 3, r = - c) c =, r = 3 d) d =8, d =

6 3º ESO Bilingüe Página General term of a geometric sequence The general term of a geometric progression is: =a r n For example, for the sequence 3, 6,, 4, 48,... The values of a and r are: a =3 (the first term) r = (the "common ratio") The general term can be calculated: =a r n =3 n =3 n Exercise. Find out the hundredth term of these arithmetic progressions: a) a =7, r = 5 b) b =6, r = 9 c) c = 5, r = 3 d) d =, r = 3.3. Sum of terms The sum of the first n terms of a geometric sequence, S n, can be calculated: S n = a r r,r For example, let's add up the first 7 terms of the geometric sequence 4,, 36,... The first term a is 4, the common ratio r is 3 and we want to add up 7 terms (n = 0). So: S n = r a r a 7 =4 3 7 =96 S 0 = = =437 Why don't you add up the terms yourself, and see if it comes to 437?

7 3º ESO Bilingüe Página 7. Find the sum of the first 00 terms of the progression:, 6, 36, 6,. Find the sum of the first 50 terms of the progression 5,, /5, /5, Find the sum of the first 0 terms of the sequence, -4, 8, -6, Sum to infinity Look at the picture and answer this question: What is the area of this square? The answer is. But then, it must be true that: =!!!! So, we are adding up infinite terms but the sum is not infinite! If r <, we can find the sum of the infinite terms of a geometric sequence, S : S= a r In the previous example, the first term a is S= a r S= and the common ratio r is = = as well, so: This time you can't be sure of your answer, but if you add up the first ten terms (for example), you can see that the sum is close to.. Find the sum to infinity of the geometric progression: 8, -7, 9,.... Find the sum to infinity of the geometric progression: 0,, 0., 0.0, 0.00, 0.000, The sum to infinity in a geometric progression is 00. Given that the first term is 5, find the common ratio, r.