Naming sequences

1. Naming sequences Name these sequences: 2, 4, 6, 8, 10, . . . Even Numbers (or multiples of 2) 1, 3, 5, 7, 9, . . . Odd numbers 3, 6, 9, 12, 15, . . . Multiples of 3 5, 10, 15, 20, 25 . . . Multiples of 5 1, 4, 9, 16, 25, . . . Square numbers 1, 3, 6, 10,15, . . . Triangular numbers

2. Objective By the end of the lesson you should be able to generate number sequences given a rule. Be able to use and generate position-to-term rules.

3. Ascending sequences ×2 +5 +5 +5 +5 +5 +5 +5 ×2 ×2 ×2 ×2 ×2 ×2 When each term in a sequence is bigger than the one before the sequence is called an ascending sequence. For example, The terms in this ascending sequence increase in equal steps by adding 5 each time. 2, 7, 12, 17, 22, 27, 32, 37, . . . The terms in this ascending sequence increase in unequal steps by starting at 0.1 and doubling each time. 0.1, 0.2, 0.4, 0.8, 1.6, 3.2, 6.4, 12.8, . . .

4. Descending sequences –7 –7 –7 –7 –7 –7 –7 –7 –1 –2 –3 –4 –5 –6 When each term in a sequence is smaller than the one before the sequence is called a descending sequence. For example, The terms in this descending sequence decrease in equal steps by starting at 24 and subtracting 7 each time. 24, 17, 10, 3, –4, –11, –18, –25, . . . The terms in this descending sequence decrease in unequal steps by starting at 100 and subtracting 1, 2, 3, … 100, 99, 97, 94, 90, 85, 79, 72, . . .

5. Sequences that increase in equal steps +4 +4 +4 +4 +4 +4 +4 We can describe sequences by finding a rule that tells us how the sequence continues. To work out a rule it is often helpful to find the difference between consecutive terms. For example, look at the difference between each term in this sequence: 3, 7, 11, 15 19, 23, 27, 31, . . . This sequence starts with 3 and increases by 4 each time. Every term in this sequence is one less than a multiple of 4.

6. Sequences that decrease in equal steps –6 –6 –6 –6 –6 –6 –6 Can you work out the next three terms in this sequence? 22, 16, 10, 4, –2, –8, –14, –20, . . . How did you work these out? This sequence starts with 22 and decreases by 6 each time. Each term in the sequence is two less than a multiple of 6. Sequences that increase or decrease in equal steps are called linear or arithmetic sequences.

7. Fibonacci-type sequences 21+13 1+1 1+2 3+5 5+8 8+13 13+21 21+34 Can you work out the next three terms in this sequence? 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, . . . How did you work these out? This sequence starts 1, 1 and each term is found by adding together the two previous terms. This sequence is called the Fibonacci sequence after the Italian mathematician who first wrote about it.

8. Describing and continuing sequences Here are some of the types of sequence you may come across: • Sequences that increase or decrease in equal steps. • These are calledlinearorarithmetic sequences. • Sequences that increase or decrease in unequal steps • by multiplying or dividing by a constant factor. • Sequences that increase or decrease in unequal steps • by adding or subtracting increasing or decreasing numbers. • Sequences that increase or decrease by adding together • the two previous terms.

9. Sequences from a term-to-term rule Write the first five terms of each sequence given the first term and the term-to-term rule. 1st term Term-to-term rule 10 Add 3 10, 13, 16, 19, 21 Subtract 5 100, 95, 90, 85, 80 100 3 Double 3, 6, 12, 24, 48 Multiply by 10 5, 50, 500, 5000, 50000 5 7 Subtract 2 7, 5, 3, 1, –1 0.8 Add 0.1 0.8, 0.9, 1.0, 1.1, 1.2

10. Writing sequences from position-to-term rules The position-to-term rule for a sequence is very useful because it allows us to work out any term in the sequence without having to work out any other terms. We can use algebraic shorthand to do this. We call the first term T(1), for Term number 1, we call the second term T(2), we call the third term T(3), . . . we call the nth term T(n). T(n) is called the the nth term or the general term.

11. Writing sequences from position-to-term rules For example, suppose the nth term of a sequence is 4n + 1. We can write this rule as: T(n) = 4n + 1 Find the first 5 terms. T(1) = 4 × 1 + 1 = 5 T(2) = 4 × 2 + 1 = 9 T(3) = 4 × 3 + 1 = 13 T(4) = 4 × 4 + 1 = 17 T(5) = 4 × 5 + 1 = 21 The first 5 terms in the sequence are: 5, 9, 13, 17 and 21.

12. Writing sequences from position-to-term rules If the nth term of a sequence is 2n2 + 3. We can write this rule as: T(n) = 2n2 + 3 Find the first 4 terms. T(1) = 2 ×12 + 3 = 5 T(2) = 2 ×22 + 3 = 11 T(3) = 2 ×32 + 3 = 21 T(4) = 2 ×42 + 3 = 35 The first 4 terms in the sequence are: 5, 11, 21, and 35. This sequence is a quadratic sequence.

13. Sequences from position-to-term rules Sometimes sequences are arranged in a table like this: We can say that each term can be found by multiplying the position of the term by 3. This is called a position-to-term rule. For this sequence we can say that the nth term is 3n, where n is a term’s position in the sequence. What is the 100th term in this sequence? 3 × 100 = 300

14. Sequences of multiples +5 +5 +5 +5 +5 +5 +5 × 5 × 5 × 5 × 5 × 5 × 5 All sequences of multiples can be generated by adding the same amount each time. They are linear sequences. For example, the sequence of multiples of 5: 5, 10, 15, 20, 25, 30 35 40 … can be found by adding 5 each time. Compare the terms in the sequence of multiples of 5 to their position in the sequence: 2 10 3 15 4 20 5 25 n Position 1 5 … … 5n Term

15. Finding the nth term of a linear sequence +3 +3 +3 +3 +3 +3 +3 × 3 × 3 × 3 × 3 × 3 + 1 + 1 + 1 × 3 + 1 + 1 + 1 The terms in this sequence 4, 7, 10, 13, 16, 19, 22, 25 … can be found by adding 3 each time. Compare the terms in the sequence to the multiples of 3. 2 3 4 5 n Position 1 … Multiples of 3 3 6 9 12 15 3n … Term 4 7 10 13 16 3n + 1 Each term is one more than a multiple of 3.

16. Finding the nth term of a linear sequence +5 +5 +5 +5 +5 +5 +5 × 5 × 5 × 5 × 5 × 5 – 4 – 4 – 4 × 5 – 4 – 4 – 4 The terms in this sequence 1, 6, 11, 16, 21, 26, 31, 36 … can be found by adding 5 each time. Compare the terms in the sequence to the multiples of 5. 2 3 4 5 n Position 1 … Multiples of 5 5 10 15 20 25 5n … Term 1 6 11 16 21 5n– 4 Each term is four less than a multiple of 5.

17. Finding the nth term of a linear sequence –2 –2 –2 –2 –2 –2 –2 × –2 × –2 × –2 × –2 × –2 + 7 + 7 + 7 × –2 + 7 + 7 + 7 The terms in this sequence 5, 3, 1, –1, –3, –5, –7, –9 … can be found by subtracting 2 each time. Compare the terms in the sequence to the multiples of –2. 2 3 4 5 n Position 1 … Multiples of –2 –2 –4 –6 –8 –10 –2n … Term 5 3 1 –1 –3 7 – 2n Each term is seven more than a multiple of –2.

18. Arithmetic sequences Sequences that increase (or decrease) in equal steps are called linear or arithmetic sequences. The difference between any two consecutive terms in an arithmetic sequence is a constant number. When we describe arithmetic sequences we call the difference between consecutive terms, d. We call the first term in an arithmetic sequence, a. For example, if an arithmetic sequence has a = 5 and d = -2, We have the sequence: 5, 3, 1, -1, -3, -5, . . .

19. The nth term of an arithmetic sequence The rule for the nth term of any arithmetic sequence is of the form: T(n) = an + b a and b can be any number, including fractions and negative numbers. For example, Generates odd numbers starting at 3. T(n) = 2n + 1 Generates even numbers starting at 6. T(n) = 2n + 4 Generates even numbers starting at –2. T(n) = 2n– 4 Generates multiples of 3 starting at 9. T(n) = 3n + 6 Generates descending integers starting at 3. T(n) = 4 – n

20. Sequences from practical contexts A possible justification of this rule is that each shape has four ‘arms’ each increasing by one tile in the next arrangement. The pattern give us multiples of 4: 1 lot of 4 2 lots of 4 3 lots of 4 4 lots of 4 The nth term is 4 ×n or 4n. Justification: This follows because the 10th term would be 10 lots of 4.