Linear number sequences
Generate and describe linear number sequences, including those with negative and decimal steps; identify the term-to-term rule
What a learner can do afterwards
- Continue the sequence 2.5, 4.0, 5.5, ... and state the rule as 'add 1.5'
- Generate terms of a sequence that crosses zero (e.g. 3, 1, −1, −3, ...)
- Describe the term-to-term rule for a given linear sequence and predict the 10th term
1 · Read
A sequence is a list of numbers written in order. In a linear sequence, you use the same rule every time to get from one number to the next.
Mia writes 2.5, 4.0, 5.5, 7.0. She checks the gaps: 4.0 minus 2.5 is 1.5, 5.5 minus 4.0 is 1.5, 7.0 minus 5.5 is 1.5. Every gap is the same, so the rule is 'add 1.5'.
Think about a cold day. The temperature is 3, then 1, then -1, then -3. Each step goes down by 2. Even when the numbers cross past zero, the rule stays the same: subtract 2.
To find a term far down the list, you do not need to write out every number. Just keep applying the rule from the term before it, again and again.
A linear sequence always uses the same add or subtract rule, even when the numbers are decimals or cross zero.
More to read
Arithmetic sequences review
A linear sequence adds the same amount each time. 2.5, 4.0, 5.5 grows by 1.5, so the next terms are 7.0 and 8.5. The rule add 1.5 never changes. Naming the rule unlocks every later term.
Recursive formulas for arithmetic sequences
Sequences can cross zero. 3, 1, -1, -3 subtracts 2 each time, sailing past zero into negatives. The 6th term is -7 and the 12th is -19. Negative terms obey the same rule as the rest.
9.2 Arithmetic Sequences
Any term follows from the start plus jumps times the gap. A sequence adding 4 from 5 reaches its 10th term as 5 + 9 x 4 = 41. Counting jumps (one fewer than the position) is the key step. The 10th term of 100, 94, 88 and so on is 100 - 9 x 6 = 46.
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2 · Watch
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Where it sits
Learn first
Optional connections
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
17 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.