Number Series & Progressions

Scenario

Finding bugs in log files often requires spotting anomalies in timestamp intervals. That's essentially sequence pattern recognition.

Mental model

Every series hides a simple engine generating the next term.

Progressions are formal math engines (constant addition or constant multiplication). Number series puzzles combine multiple basic engines, like squares, primes, and alternating steps.

Formula

AP: Nth term = a + (n-1)d. Sum of N terms = n/2 x (2a + (n-1)d) = n/2 x (first + last). GP: Nth term = a x r^(n-1). Sum of N terms = a x (r^n - 1) / (r - 1) for r != 1. Sum of infinite GP = a / (1 - r) for |r| < 1. Special sums: sum of first N naturals = N(N+1)/2. Sum of first N squares = N(N+1)(2N+1)/6. Sum of first N cubes = [N(N+1)/2]^2. AP average = (first + last) / 2 = middle term (if odd count).

AP grows by a constant difference d (linear). GP grows by a constant ratio r (exponential). For series puzzles, always compute first differences, then second differences - most OA series reveal their pattern by the second tier.

Pattern discovery

Find the differences between consecutive terms and observe the second-level differences.

Series: 2, 5, 10, 17, 26
First differences: 3, 5, 7, 9 (odd numbers, constant gap of 2)
Second differences: 2, 2, 2 (constant -> quadratic series)

If first differences form an arithmetic progression, what type of series is the original?

Rule: If the first differences form an arithmetic progression, the original series is quadratic (squares). Constant first differences = linear (AP). Constant ratio of terms = geometric (GP).

Worked example

Problem: Find the 15th term and the sum of the first 15 terms of the AP: 3, 7, 11, 15, ...

Identify: a = 3, d = 7 - 3 = 4.
15th term = a + (n-1)d = 3 + (15-1) x 4 = 3 + 56 = 59.
Sum of 15 terms = n/2 x (first + last) = 15/2 x (3 + 59) = 7.5 x 62 = 465.
Sanity-check using formula: n/2 x (2a + (n-1)d) = 15/2 x (6 + 56) = 7.5 x 62 = 465. Matches.

Answer: 15th term = 59, Sum = 465.

Problem: What is the missing term in the series: 4, 9, 19, 34, 54, ?

Find first differences: 9-4=5, 19-9=10, 34-19=15, 54-34=20.
First differences: 5, 10, 15, 20 - these increase by 5 each time (AP with d=5).
Next first difference = 25.
Missing term = 54 + 25 = 79.
Sanity-check: The pattern of differences (5, 10, 15, 20, 25) is consistent.

Answer: 79.

Shortcuts

Double Difference Method: If a pattern is not obvious, calculate the difference between terms, then the difference between those differences. Most OA series break down by the second tier. (1, 4, 10, 19. Diffs: 3, 6, 9. Second diffs: 3, 3. Next diff = 12, so next term = 19+12 = 31.)
The Median Average for AP: The average of an AP is always exactly its middle term (if n is odd), or the average of its first and last terms. (Average of 2, 4, 6, 8, 10 is 6. Also (2+10)/2 = 6.)

Common mistakes

Assuming constant ratio in small samples: If a series is 2, 4..., the next term could be 8 (GP), or 6 (AP), or 7 (primes). Always confirm the pattern with at least 3 transitions.
Miscounting the N in progressions: The number of terms between x and y inclusive is ((y - x) / d) + 1. Forgetting the '+1' is the most common off-by-one error.

Glossary

engine: A simple hidden rule or formula that generates the next part of a pattern.
exponential: Growing very quickly by multiplying, rather than just adding.
inclusive: Counting both the starting number and the ending number, as well as everything in between.

Recall questions

What is the formula for the nth term of a Geometric Progression?
What is the sum of the first N natural numbers?
If an AP has a common difference of 0, what is it?

Questions & answers

Find the missing term: 3, 7, 16, 32, 57, ?

Approach: Differences: 4, 9, 16, 25. These are squares (2^2, 3^2, 4^2, 5^2). Next difference is 6^2 = 36. 57 + 36 = 93.

Find the sum of all 2-digit numbers divisible by 3.

1665

Approach: AP: first term a=12, last term L=99, d=3. N = ((99-12)/3) + 1 = 30. Sum = 30/2 x (12 + 99) = 15 x 111 = 1665.

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