These questions ask you to continue a sequence, find its signed step, fill a gap, count moves, work backwards or identify the first term on the other side of zero. Some patterns add a fixed integer, while others multiply by a fixed integer.
Find the rule before extending the pattern
For an arithmetic sequence, subtract each term from the term immediately after it. The result must be the same signed common difference across at least two gaps. In 8, 3, -2, the calculation 3 minus 8 gives -5 and -2 minus 3 also gives -5, so every forward move adds -5. Zero does not stop or reverse the rule; a step can land on zero or pass straight over it.
| Task | Decision procedure |
|---|---|
| Next arithmetic term | Add the signed common difference once. |
| Earlier term | Undo the common difference by using its opposite. |
| Term in a stated position | From the first term, apply the step one fewer times than the position number. |
| One missing middle term | Split the total change between the known neighbours into two equal signed steps. |
| Repeated multiplication | Divide consecutive nonzero terms to identify the common integer multiplier, then multiply again. |
To move backwards, reverse the operation rather than merely reading the same step in the other direction. If the forward step is +6, the preceding term is found by subtracting 6; the term before 1 would be -5. For position questions, count gaps, not written values. The fourth term lies three moves after the first. For a multiplicative pattern such as -3, 6, -12, the constant multiplier is -2, so signs alternate and magnitudes double.
Sequence traps to avoid
- Using an unsigned gap for a falling sequence loses the direction. Compute later term minus earlier term so the common difference carries its minus sign.
- Counting the first term as the first move creates an off-by-one result. A term in position five is four equal gaps beyond the start.
- Applying the forward step while moving backwards finds the following term instead of the preceding one. Reverse addition to subtraction, or subtraction to addition.
- Counting displayed values instead of moves overstates the number of steps by one. Mark the gaps between consecutive terms.
- Assuming every sequence must land on zero invents a term that the constant step may skip. Apply the step unchanged and identify the first result that is actually negative.
- Treating alternating signs as repeated addition misses a constant negative multiplier. Compare ratios as well as differences before deciding the pattern family.
Try it
A sequence runs -3, -8, -13 and continues with the same step. Enter the next term.
A sequence has equal steps and runs -11, then one missing term, then -3. What is the missing term?
- -14
- -15
- 7
- -7
A sequence runs -4, -9, -14 and continues with the same step. Enter its sixth term.