When a sequence does not increase by a fixed amount, its increases may have a pattern of their own. A difference table makes that structure easier to see.
Calculate the first differences
Take 3, 8, 15, 24, 35. Subtract each term from the following term. The differences are 5, 7, 9 and 11.
These are not constant, but they rise by two. Those new differences—2, 2 and 2—are the second differences. Continuing the first-difference pattern gives 13, so the next sequence term is 35 + 13 = 48.
| Layer | Values |
|---|---|
| Original sequence | 3, 8, 15, 24, 35 |
| First differences | 5, 7, 9, 11 |
| Second differences | 2, 2, 2 |
Predict the increment before the term
A frequent mistake is to add the second difference directly to the last original term. Here, adding two to 35 would give 37, which skips a level of the table.
Instead, update the last first difference: 11 + 2 = 13. Then update the original sequence: 35 + 13 = 48. Keep those two operations separate.
Practice with a decreasing increment
For 50, 66, 79, 89, the increases are 16, 13 and 10. Continuing the change of minus three gives an increase of seven, so the next term is 96.
The sequence is still increasing even though its increments are decreasing. Read the layer you are describing carefully.
Know when to change approach
If the difference table becomes more complicated at each level, try ratios, alternation or interleaved positions. You can construct formulas for a finite set of terms, but that is not always the intended reasoning task. Seek a compact explanation that accounts for the available evidence, then verify it against every term and the answer choices.