5.2 Number Series: Differences, Ratios, and Structured Tests

Read the displayed sequence

Riverside describes Number Series as recognizing a rule in a series of numbers and selecting the number that comes next. Unlike Number Analogies, this format presents a sequence to extend. The original examples here teach ways to compare neighboring terms and test a candidate throughout the sequence.

A finite sequence can support many mathematical extensions. In practice, use a straightforward rule that accounts for the displayed transitions and is supported by the options. Do not claim that a few terms prove a unique formula among all imaginable functions.

Test a constant difference

An arithmetic progression adds the same difference each time. For $7,11,15,19$, the differences are four, four, and four, so the simple continuation is twenty-three.

Compute all displayed differences, not just the first. For $7,11,16,22$, the differences are four, five, and six. Adding four again would ignore the later evidence. A rule may be valid for one transition and fail the full sequence.

The third and fourth examples require a pattern in the differences, rather than a constant difference. The table illustrates methods; it does not publish official frequencies for these series types.

Test a constant ratio

A geometric progression multiplies by the same factor. For $3,6,12,24$, the ratio is two throughout, and the next term under that rule is forty-eight. For $81,27,9,3$, division by three gives one next.

Ratios can be fractional or negative. The sequence $16,8,4,2$ uses a factor of one-half. The sequence $2,-6,18,-54$ uses a factor of negative three and predicts positive one hundred sixty-two.

Check zero before dividing. If a term is zero, the ratio to it is undefined; a difference method or another rule may be more appropriate. Do not try to divide by zero to force a ratio comparison.

Compare first and second differences

For $2,5,10,17,26$, the first differences are three, five, seven, and nine. The second differences are two, two, and two. Continuing that pattern gives a next first difference of eleven and a next term of thirty-seven.

These terms also match $n^2+1$ starting with $n=1$. A direct formula and a difference description can represent the same pattern. You do not need to derive the formula if extending the differences is sufficient, but both routes can verify the calculation.

Constant second differences suggest a quadratic pattern when terms are indexed at equal steps. They do not prove that the test owner intended a particular formula or that no alternative finite-data extension exists. Keep the claim proportional to the evidence.

Work a compound transition

For $2,5,11,23,47$, try “double the previous term, then add one.” Check each step:

• $2(2)+1=5$.
• $2(5)+1=11$.
• $2(11)+1=23$.
• $2(23)+1=47$.

The next term is $2(47)+1=95$. Adding the last difference of twenty-four would give seventy-one, but the differences are themselves doubling rather than constant. A complete check distinguishes the two hypotheses.

A rule based on a prior term is called a recurrence. It can use one prior term, as this example does, or more than one. Recurrence does not always mean adding the two preceding numbers.

Use bounds to detect arithmetic errors

If every transition multiplies a positive number by two and adds one, each output should be more than double the previous term. A predicted next term smaller than ninety-four in the last example fails that bound. This does not replace the calculation, but it can catch a copying or addition error.

For repeated halving of positive numbers, the next term must be smaller but remain positive. For repeated subtraction, the terms may eventually become negative. Use the candidate rule's actual consequences, not a blanket belief that a series must increase or remain integral.

Choose the next test deliberately

Begin by asking whether differences are constant. If they are not, inspect whether ratios are constant, whether differences have a simple pattern, or whether transitions alternate. Avoid running every possible method on every item. The visible numbers guide a sensible next check.

When a candidate fits, calculate the next term and verify the last displayed transition again. A sign error in the last difference can otherwise carry directly into your answer. If two candidates remain plausible and predict different available answers, inspect any additional wording before deciding.

Learn from a missed continuation

Record the full sequence, the candidate rule, and the first transition where it fails. “Added the final difference” is useful only when you also note why that difference was not constant. Solve a fresh series with the same structure to check transfer.

The complete standard Number Series subtest has 18 items in 10 minutes. Use optional timed practice after these checks become reliable. Scratch paper is permitted under current proctoring guidance; calculators are not. Neither a fast answer nor an elegant formula compensates for a rule that fails a displayed transition.