Number & letter series
Mental model
Every series is driven by a hidden rate of change.
Slowly growing series are additive, rapidly growing series are multiplicative/exponential, and erratic series are actually two interwoven series.
Method
- Examine the overall rate of growth between terms to identify the likely mathematical driver (addition for slow growth, multiplication/squares for rapid growth).
- Write down the first-level differences (the gap between consecutive terms) to expose hidden additive patterns.
- If the first-level differences don't reveal a clear pattern, calculate the second-level differences (the gap between the gaps).
- For letter series, immediately convert all letters to their alphabetical numerical equivalents (A=1, B=2 ... Z=26) and treat it as a standard number series.
- Apply the validated mathematical operation or difference pattern to the last known term to compute the missing term.
Worked example
Problem: Find the next number in the series: 2, 5, 11, 23, 47, ?
- Check rate of growth: The numbers roughly double each step, suggesting a multiplicative rule or rapidly growing additive rule.
- Find first-level differences: 5-2=3, 11-5=6, 23-11=12, 47-23=24.
- Analyze differences: The sequence of gaps is 3, 6, 12, 24. These differences are doubling at each step.
- Predict next difference: The next gap should be 24 * 2 = 48.
- Apply to last known term: Add the next gap to the last term: 47 + 48 = 95.
Answer: 95
Common mistakes
- Not calculating the second-level differences: Staring at the first level of differences without realizing that the pattern only becomes obvious when you subtract the differences from each other.
- Trying to solve letter series without numbers: Counting letters on your fingers is slow and prone to errors. Always convert letters to numeric values immediately.
Glossary
- multiplicative
- Using multiplication to increase or decrease numbers.
- interwoven
- Mixed together so that two different patterns look like a single complicated one.
- additive
- Using addition to increase numbers step-by-step.
Recall questions
- What is the first step you should take when analyzing a new number series?
- What should you do if the first-level differences between terms do not show an obvious pattern?
- What is the most efficient way to approach a letter series problem?
Questions & answers
Find the missing term: B, E, H, K, ?
N
Approach: Convert to numbers: 2, 5, 8, 11. Difference is +3. Next is 14, which maps to N.
Find the next number: 1, 4, 9, 16, 25, ?
36
Approach: Recognize the direct pattern of perfect squares: 1^2, 2^2, 3^2, 4^2, 5^2. Next is 6^2 = 36.