Mastering Sequences in Mathematics
Delve into the world of mathematical sequences with this comprehensive guide. Explore what sequences are, different types like arithmetic and geometric, rules for continuing sequences, methods to find specific terms, and understanding position-to-term rules to solve for nth terms efficiently.
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Year 8 Lockdown lessons
Week 6: Lesson 1 Introduction to sequences
What is a sequence? A mathematical sequence is a series of numbers connected by a rule. This rule defines their pattern 3, 8, 13, 18, 23 +5 every time 2, 4, 8, 16, 32 x2 every time -4 every time 11, 7, 3, -1, -5 Each part of a sequence is called a term How we move from one term to the next is called the term to term rule.
Types of sequence There are two key types of sequences. Arithmetic sequences These have a term to term rule which is either an addition or a subtraction. Geometric sequences These have a term to term rule which is either an multiplication or a division.
Continuing sequences For each sequence below.. a) Find the term to term rule b) State if it is arithmetic or geometric c) Find the next 2 terms x2 -1 +5 +2 Geometric 80, 160 Arithmetic 16, 15 Arithmetic 25, 30 Arithmetic 11, 13 +4 -6 +0.5 x10 Arithmetic 26, 30 Arithmetic -9, -15 Arithmetic 4, 4.5 Geometric 10000, 100000
Finding specific terms What is the 7th term in this sequence? 2, 5, 8, 11 +3 +3 +3 +314 +317 +320 What is the 70th term in this sequence? We need a better method than just moving along the sequence!
Finding specific terms Fortunately all sequences also have a position to term rule 3n 1 2, 5, 8, 11, 14, 17, 20 The position we are looking for is n So the 7thterm is = 20 3 x 7 - 1 And the 70thterm is = 209 3 x 70 - 1 We often call this the nth term rule.
Finding specific terms 1st term 2nd term 3rd term 10th term 50th term Rule 5 x 50 + 1 = 251 5 x 10 + 1 = 5 x 2 + 1 = 5 x 3 + 1 = 5 x 1 + 1 = 5n + 1 51 11 16 6 30 2 x 50 = -70 30 2 x 10 = 10 30 2 x 2 = 30 2 x 3 = 30 2 x 1 = 30 2n 26 24 28 6 x 50 3 = 297 6 x 10 3 = 6 x 2 3 = 6 x 3 3 = 6 x 1 3 = 6n 3 57 9 15 3 50 + 8 = 10 + 8 = 2 + 8 = 3 + 8 = 1 + 8 = n + 8 58 18 10 11 9
Finding specific terms 3 x 4 + 6 = 18 Q1) What is the 4th term in the sequence 3n + 6? 2 x 8 3 = 15 Q2) What is the 8th term in the sequence 2n 3? 50 2 x 15 = 20 Q3) What is the 15th term in the sequence 50 2n? 4 x 37 1 = 147 Q4) What is the 37th term in the sequence 4n 1? 10 x 102 15 = 1005 Q5) What is the 102nd term in the sequence 10n 15? Q6) Find the first 4 terms in each of the sequences below. e) 30 4n c) n + 11 d) 2n 7 a) 4n 3 b) 7n + 1 26, 22, 18, 14 1, 5, 9, 13 8, 15, 22, 29 12, 13, 14, 15 -5, -3, -1, 1
Checking if a number is in a sequence Is 101 in the sequence 2n+1? Yes, it is the 50th term in this sequence Position (n) Value of the term (the sequence) x2 +1 50 101 2 -1
Checking if a number is in a sequence Is 30 in the sequence 2n+1? No, it is between the 14th and 15th terms Position (n) Value of the term (the sequence) x2 +1 14.5 30 2 -1
Is 57 in the sequence 4n + 5? Is 84 in the sequence 3n + 2? 13 4 -5 57 27.333 3 -2 84 Yes, it is in the sequence No, it is not in the sequence Is 107 in the sequence 8n 3? Is 84 in the sequence 5n 1? 13.75 8 +3 107 17 5 +1 84 No, it is not in the sequence Yes, it is in the sequence
Checking if a number is in a sequence 1) Is 73 in the sequence 5n 2? +2 5 Yes, it is the 15th term in the sequence = 15 2) Is 27 in the sequence 4n + 1? -1 4 No, it is between the 6th and 7th terms = 6.5 3) Is 71 in the sequence 3n + 8 ? -8 3 Yes, it is the 21st term in the sequence = 21 4) Is 98 in the sequence 7n - 5? +5 7 No, it is between the 14th and 15th terms = 14.714 5) What is the first number in the sequence 3n + 2 to be over 250? (250 2) 3 = 82.66 The 83rd term will be the first one that is over 250. 3 x 83 + 2 = 251 6) What is the first number in the sequence 8n 5 to be over 100? The 14th term will be the first one that is over 100. (100 + 5) 8 = 13.125 8 x 14 5 = 107
Year 8 Lockdown lessons
Challenge 1 2 x 5 - 4 6 x5 -4 26 4 22 2 x - 6 -2 2 x4 a = 4, b = 2
Challenge 2 We need to work out what position in the sequence we would find 0 150 4n = 0 0 is between term 37 and term 38. 150 = 4n n = 37.5 Term 38 will therefore be the first negative term 150 4 x 38 = -2
Challenge 3 = 8 = 2 x 4 1st term (n=1): (1 + 1)(1 + 3) = 15 = 3 x 5 2nd term (n=2): (2 + 1)(2 + 3) = 24 = 4 x 6 3rd term (n=3): (3 + 1)(3 + 3) = 8, 15, 24
