6.1 Number Series (Mathematical Sequences)

Number Series on the OLSAT Level G

Number series questions are a staple of the Quantitative Reasoning section of the OLSAT Level G. These questions present you with a sequence of numbers that follow a specific mathematical rule or logical pattern. Your task is to determine the rule governing the sequence and apply it to find the missing number, which is typically the next number in the series, though occasionally it may be in the middle.

Success in this area relies on your ability to quickly test hypotheses about mathematical relationships. You must be comfortable with basic arithmetic operations—addition, subtraction, multiplication, and division—and be able to recognize when these operations are combined in complex ways. In this section, we will explore the most common types of number series you will encounter on the exam, breaking down the strategies for identifying and solving each type.

Arithmetic Progressions

The most fundamental type of number series is the arithmetic progression. In an arithmetic series, the difference between any two consecutive numbers is constant. This means the pattern involves adding or subtracting the same value at each step.

Simple Addition and Subtraction

A simple addition series might look like this: 2, 5, 8, 11, 14, __. To solve this, find the difference between consecutive terms:

• 5 - 2 = 3
• 8 - 5 = 3
• 11 - 8 = 3
• 14 - 11 = 3

The rule is clearly "add 3." Therefore, the next number is 14 + 3 = 17.

Similarly, a subtraction series involves a constant decrease: 50, 45, 40, 35, 30, __.

• 45 - 50 = -5
• 40 - 45 = -5

The rule is "subtract 5," making the next number 30 - 5 = 25.

Increasing or Decreasing Differences

Sometimes, the difference itself forms a pattern. Consider the series: 2, 4, 7, 11, 16, __. Let's find the differences:

• 4 - 2 = 2
• 7 - 4 = 3
• 11 - 7 = 4
• 16 - 11 = 5

The differences are not constant, but they are increasing by 1 each time (+2, +3, +4, +5). The next difference must be +6. Therefore, the next number is 16 + 6 = 22. This is a "second-order" arithmetic series, where the differences between the numbers form a simple arithmetic series.

Geometric Progressions

In a geometric progression, each term is found by multiplying or dividing the previous term by a constant number. These sequences grow or shrink much faster than arithmetic progressions.

Multiplication Series

Consider the sequence: 3, 6, 12, 24, 48, __. If we look at the differences, we get +3, +6, +12, +24. While the differences double, it's easier to look at the ratio between terms:

• 6 / 3 = 2
• 12 / 6 = 2
• 24 / 12 = 2
• 48 / 24 = 2

The rule is "multiply by 2." The next number is 48 * 2 = 96. When you see a sequence that increases rapidly, always check for a multiplication rule first.

Division Series

A division series shrinks rapidly: 162, 54, 18, 6, __. Check the ratios:

• 54 / 162 = 1/3 (or divided by 3)
• 18 / 54 = 1/3
• 6 / 18 = 1/3

The rule is "divide by 3." The next number is 6 / 3 = 2.

Multi-Step Operations

The OLSAT Level G will frequently test your ability to recognize patterns that involve more than one operation at a time. These multi-step rules can be challenging because a simple difference or ratio check won't yield a constant value.

Multiply and Add/Subtract

Consider the series: 2, 5, 11, 23, 47, __. The differences are +3, +6, +12, +24. You might recognize that the differences are doubling. So the next difference is +48, making the answer 47 + 48 = 95.

However, there is another way to view this sequence:

• 2 * 2 + 1 = 5
• 5 * 2 + 1 = 11
• 11 * 2 + 1 = 23
• 23 * 2 + 1 = 47

The rule "multiply by 2, then add 1" also yields the next number: 47 * 2 + 1 = 95. Recognizing these compound rules is a critical skill for advanced number series.

Alternating Operations

Sometimes the operations alternate between steps. For example: 10, 50, 45, 225, 220, __. Let's analyze the steps:

• 10 to 50: Multiply by 5
• 50 to 45: Subtract 5
• 45 to 225: Multiply by 5
• 225 to 220: Subtract 5

The pattern alternates between "multiply by 5" and "subtract 5." Since the last operation was subtract 5, the next operation must be multiply by 5. The next number is 220 * 5 = 1100.

Alternating / Interleaved Series

An alternating or interleaved series is actually two separate sequences woven together. If a sequence seems to jump up and down erratically, or if the standard rules don't seem to apply, try looking at every other number.

Consider the series: 3, 20, 6, 18, 9, 16, 12, __. The differences are +17, -14, +12, -9, +7, -4. This doesn't seem to follow a clear, simple rule.

Now, let's split the series into two interleaved sequences:

• Sequence A (odd positions): 3, 6, 9, 12...
• Sequence B (even positions): 20, 18, 16...

Analyze them separately:

• Sequence A rule: Add 3 (+3)
• Sequence B rule: Subtract 2 (-2)

The blank is in the 8th position, which belongs to Sequence B. The last number in Sequence B was 16. Applying the rule "subtract 2," we get 16 - 2 = 14. The missing number is 14.

Fibonacci-Like Series

In a Fibonacci sequence, each number is the sum of the two preceding numbers. The classic Fibonacci sequence is: 1, 1, 2, 3, 5, 8, 13, 21... (where 1+1=2, 1+2=3, 2+3=5, etc.).

On the OLSAT, you may encounter variations of this logic. The rule "add the previous two numbers" might be modified. Example: 2, 4, 6, 10, 16, 26, __.

• 2 + 4 = 6
• 4 + 6 = 10
• 6 + 10 = 16
• 10 + 16 = 26 To find the next number, add the two preceding numbers: 16 + 26 = 42.

You might even see combinations, such as "add the previous two numbers and then add 1." Example: 1, 2, 4, 7, 12, __.

• (1 + 2) + 1 = 4
• (2 + 4) + 1 = 7
• (4 + 7) + 1 = 12 The next number would be (7 + 12) + 1 = 20.

Strategies for Number Series Success

1. Calculate the Differences: Your first step should always be to write out the differences between consecutive terms. This will immediately reveal simple arithmetic progressions and will often expose secondary patterns (like increasing differences).
2. Check the Growth Rate: If the numbers are getting very large very quickly, addition is unlikely to be the primary rule. Start looking for multiplication or exponents. Conversely, if the numbers shrink rapidly, look for division.
3. Look for the "See-Saw" Pattern: If the numbers alternate between going up and going down (e.g., 5, 10, 8, 16, 14), you are likely dealing with either alternating operations (like multiply then subtract) or an interleaved series.
4. Test Your Rule: Once you formulate a hypothesis for the rule, test it on every single number provided in the sequence. A common trap is a rule that works for the first three numbers but fails on the fourth. The correct rule must apply flawlessly to the entire given sequence.
5. Memorize Common Squares and Cubes: Recognizing perfect squares (1, 4, 9, 16, 25, 36...) and perfect cubes (1, 8, 27, 64, 125...) can save you immense amounts of time, as these frequently form the basis of advanced sequences.