Equivalent Expressions and Number Sequences
Primary 6 · Mathematics
Equivalent Expressions and Number Sequences
Lesson 5 of 5
15 min
Let's begin
Chairs are set out in the hall for a school meeting. Row 1 has 5 chairs, row 2 has 8, row 3 has 11, and so on. The head teacher asks Byiringiro how many chairs will be in row 20. Byiringiro does not want to count all 20 rows. With a general rule, he doesn't have to.
Put the sequence in a table. The position (n) tells you where the term is.
| Position n | 1 | 2 | 3 | 4 | … | 20 |
|---|---|---|---|---|---|---|
| Term | 5 | 8 | 11 | 14 | … | ? |
A sequence with a common difference is called a linear sequence. Here the common difference is 3.
Step 1 the common difference goes with n. The difference is 3, so start with 3n.
Step 2: compare 3n with the real terms. When n = 1, 3n = 3, but the term is 5. We need 2 more.
Step 3: the rule is 3n + 2. Check: n = 2 gives 3 × 2 + 2 = 8, and n = 3 gives 11. Correct.
Row 20: 3 × 20 + 2 = 62 chairs.
Another way: the nth term = first term + (n − 1) × difference. For row 20: 5 + 19 × 3 = 5 + 57 = 62. Both ways agree.
Worked example 1
7, 11, 15, 19, … The difference is 4, so start with 4n. When n = 1, 4n = 4, but the term is 7, so add 3. Rule: 4n + 3. The 50th term is 4 × 50 + 3 = 203.
Worked example 2
(going down). 40, 37, 34, … Each term is 3 less. Using the other way: term = 40 + (n − 1) × ( − 3) = 43 − 3n. The 10th term is 43 − 30 = 13.
Worked example 3
(making a sequence from a rule). The rule 2n − 1 gives 2 × 1 − 1 = 1, then 3, 5, 7, 9: the odd numbers.
Worked example 4
(finding the position). Which term of 5, 8, 11, … is 65? Set the rule equal to 65: 3n + 2 = 65, so 3n = 63 and n = 21. It is the 21st term.
Worked example 5
(fractions and decimals). 12, 1, 1 12, 2, … has difference 12, and the rule is n ÷ 2. The 9th term is 9 ÷ 2 = 4 12.
0.5, 0.8, 1.1, … has difference 0.3. Rule: 0.3n + 0.2. The 10th term is 3 + 0.2 = 3.2.
What is 1 + 2 + 3 + … + 10? Pair the numbers from the two ends: 1 + 10 = 11, 2 + 9 = 11, 3 + 8 = 11, 4 + 7 = 11, 5 + 6 = 11. That is 5 pairs of 11, so the total is 55. In the same way, 1 + 2 + … + 20 is 10 pairs of 21, which is 210.
Tip
The rule of a linear sequence is difference × n + adjustment. The adjustment is the first term minus the difference.
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What is the general rule of the sequence 6, 10, 14, 18, …?
