MATHEMATICSNumber Sequences – Finding next or missing terms

Aims of the Lesson

• To investigate linear number sequences.

• To learn how to find the next terms of a linear sequence.

• To learn how to find previous or missing terms in a linear sequence.

Example• Make chains of matchstick squares and count the number of matches needed

4 7 10

In this sequence we are adding 3 more matches each time, so we are adding 3 on to the last number each time. [Adding 3 each time will give us a rule connected to the 3 times table.]

Describing Rules

• There are two ways of describing a rule for sequences…

either relating one term to the next term

or relating a term to its position in the sequence

• For the matchstick example…

The term-to-term rule is ADD 3 (to the last term)

The position-to-term rule is MULTIPLY (the position) by 3 then add 1 on

Describing Rules (cont’d)

• Term-to-term rules are usually described in words (add 3) or using operations and numbers (+3)

• Position-to-term rules can be described in words (multiply by 3 then add 1) but are more usually expressed algebraically (i.e. 3n+1)

Arithmetic Sequences

• First find the common difference – the difference between two consecutive terms (which ONLY works for linear arithmetic sequences).

• Use this to find the next or missing values

• Remember that if you need to find earlier values than the one shown, you need to do the opposite operation.

• E.g. Add 3 going forward becomes subtract 3 going backwards.

Examples• Find the next 2 terms and the rule for: 2, 5, 8, 11, 14 ….

2 5 gives us the rule of:[CHECK: 5 8 is also +3]

Number after 14 14 + 3 =Number after 17 17 + 3 =

• Find the missing terms and the rule for: 5, __, 19, 26, __

19 26 gives us the rule of:Number before 19 19 – 7 =[CHECK: 5 + 7 also gives 12]Number after 26 26 + 7 =

Add 3 (common diff = +3)

Add 7 (common diff = +7)

1720

12

33

Harder Examples

• Find the missing terms and rule for: ___, 27, ___, 19, 15

19 15 gives us the rule of:Number after 27 27 – 4 =[CHECK: 23 23 – 4 = 19!]

Number before 27 27 + 4 =

• Find the missing terms and rule for: 48, ___, 70 , ___, 92

48 70 (2 jumps!) gives us: Add 22So our rule for one jump is half this Number after 48 48 + 11=[CHECK: 59 59 + 11 = 70!]

Number after 70 70 + 11 =

Take 4 (common diff = –4)

Add 11 (common diff = +11)

23

59

31

81

REMEMBER

Please remember the following:• Always check answers in another way

• Use an arrow not an equal sign if the statements on either side are not equal

• E.g. 19 = 19 + 3

• Should be: 19 19 + 3 = 22• This shows that workings using the 19 were 19 + 3 and that these were equal

to 22!

(is wrong because 19 does NOT equal 22!)

What next?

Make notes (including examples) on finding the next or missing terms in a linear arithmetic sequence.

Work through the MyMaths lesson (and the its online homework) called:Algebra > Sequences > Arithmetic Sequences

Save and complete the worksheet: LinSeq-S1.xlsx

Now move on the Seq-Nth powerpoint…