Number sequences, terms, the general term, terminology. Sequences and Series Number sequences, terms, the general term, terminology.

Formulas booklet page 3

Each number in a sequence is called a term. In maths, we call a list of numbers in order a sequence. Each number in a sequence is called a term. 4, 8, 12, 16, 20, 24, 28, 32, . . . 1st term 6th term

Infinite and finite sequences A sequence can be infinite. That means it continues forever. For example, the sequence of multiples of 10, 10, 20 ,30, 40, 50, 60, 70, 80, 90 . . . is infinite. We show this by adding three dots at the end. If a sequence has a fixed number of terms it is called a finite sequence. Pupils may ask the difference between a sequence of numbers and a number pattern. A sequence, unlike a number pattern, does not need to follow a rule or pattern. It can follow an irregular pattern, affected by different factors (e.g. the maximum temperature each day); Or consist of a random set of numbers (e.g. numbers in the lottery draw). These ideas can be discussed in more detail during the plenary session at the end of the lesson. For example, the sequence of two-digit square numbers 16, 25 ,36, 49, 64, 81 is finite.

Infinite and finite sequences A sequence can be infinite. That means it continues forever. For example, the sequence of multiples of 10, 10, 20 ,30, 40, 50, 60, 70, 80, 90 . . . is infinite. We show this by adding three dots at the end. If a sequence has a fixed number of terms it is called a finite sequence. Pupils may ask the difference between a sequence of numbers and a number pattern. A sequence, unlike a number pattern, does not need to follow a rule or pattern. It can follow an irregular pattern, affected by different factors (e.g. the maximum temperature each day); Or consist of a random set of numbers (e.g. numbers in the lottery draw). These ideas can be discussed in more detail during the plenary session at the end of the lesson. For example, the sequence of two-digit square numbers 16, 25 ,36, 49, 64, 81 is finite.

Naming sequences Here are the names of some sequences which you may know already: 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 This can be done as an oral activity. It may also be useful for pupils to copy these number patterns into their books. As each number sequence is revealed ask the name of the sequence before revealing it. Year 7 pupils should have met square numbers and triangular numbers in Year 6. They will be revisited in more detail later in the year. Pupils may describe the pattern of square numbers either as adding 3, adding 5, adding 7 etc. (i.e. adding consecutive odd numbers) or as 1 × 1, 2 × 2, 3 × 3, 4 × 4, 5 × 5 etc. Pupils may need help verbalizing a rule to generate triangular numbers (i.e. add together consecutive whole numbers). 1, 4, 9, 16, 25, . . . Square numbers 1, 3, 6, 10,15, . . . Triangular numbers

Ascending sequences 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, . . . +5 +5 +5 +5 +5 +5 +5 Stress the difference between sequences that increase in equal steps (linear sequences) and sequences that increase in unequal steps (non-linear) sequences. 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, . . . ×2 ×2 ×2 ×2 ×2 ×2 ×2

Descending sequences 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, . . . –7 –7 –7 –7 –7 –7 –7 Stress the difference between sequences that decrease in equal steps (linear sequences) and sequences that decrease in unequal steps (non-linear) sequences. 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, . . . –1 –2 –3 –4 –5 –6 –7

Describe the following number patterns and write down the next 3 terms: Add 3 15, 18, 21 Multiply by -2 48, -96, 192 On your calculator type 3, enter, times -2, enter, keep pressing enter to generate next terms.

The general term of a sequence.

Generate the first 4 terms of each of the following sequences.

Using your calculator:

Defining a sequence recursively. Example: Find the first four terms of each of the following sequences

Using TI Nspire In a Graph screen select Sequence and enter as shown. Ctrl T to see a table of values

Defining a sequence recursively. Fibonacci Sequence.

Series and Sigma notation. When we add the terms of a sequence we create a series. sequence series

Sigma notation (summation)

Write in an expanded form first then use GDC to evaluate.

Ex 7A and & 7B from the handout in SEQTA Exercise 8.1.3 p 268 Questions 12, 13