Section 3.3: Nonlinear Sequences

Nonlinear Sequences

Nonlinear sequences represent a significant step up in difficulty on the EDPT. In these sequences, the rule itself is changing as you move through the sequence. There is no single common difference and no single common ratio. Instead, the relationship between the numbers evolves dynamically.

Mastering nonlinear sequences requires you to memorize certain number sets (like squares and cubes) and to deploy advanced analytical techniques like "second differences."

Perfect Squares and Cubes

The most common nonlinear sequences rely on exponents. If you do not have the first 12 perfect squares and the first 5 perfect cubes memorized, you will lose precious time calculating them on the exam.

Perfect Squares

The sequence of perfect squares is: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144...

• Rule: $n^2$ where $n = 1, 2, 3...$

Test-makers often disguise squares by adding or subtracting a constant. Consider the sequence: 2, 5, 10, 17, 26, ...

• Notice how close these are to the squares?
• 2 is (1 + 1)
• 5 is (4 + 1)
• 10 is (9 + 1)
• The rule is $n^2 + 1$. The next term is $36 + 1 = 37$.

Perfect Cubes

The sequence of perfect cubes is: 1, 8, 27, 64, 125, 216...

• Rule: $n^3$ where $n = 1, 2, 3...$

Like squares, cubes can be modified. Consider: 0, 7, 26, 63, 124, ...

• This is simply $n^3 - 1$. The next term is $216 - 1 = 215$.

The Fibonacci Sequence and Additive Patterns

The Fibonacci sequence is a famous mathematical pattern where each number is the sum of the two preceding numbers.

1, 1, 2, 3, 5, 8, 13, 21, 34...

• 1 + 1 = 2
• 1 + 2 = 3
• 2 + 3 = 5

The EDPT will almost never give you the standard Fibonacci sequence, as it is too recognizable. Instead, they will use Fibonacci-style logic starting with different numbers.

Consider: 4, 7, 11, 18, 29, 47, ...

• 4 + 7 = 11
• 7 + 11 = 18
• 11 + 18 = 29
• 18 + 29 = 47
• The next term is $29 + 47 = 76$.

When a sequence grows steadily, but you cannot find a common difference, ratio, or multi-step pattern, always check if the terms are being added to one another.

Pronic Sequences and Quadratic Variations ($n^2 \pm n$)

Pronic numbers (also known as rectangular numbers) are numbers that are the product of two consecutive integers: $n(n + 1)$. This can also be expressed as $n^2 + n$. The canonical pronic sequence is: 2, 6, 12, 20, 30, 42, 56, 72, 90, 110...

• 1 * 2 = 2
• 2 * 3 = 6
• 3 * 4 = 12
• 4 * 5 = 20

A closely related sequence is $n(n - 1)$ or $n^2 - n$: 0, 2, 6, 12, 20, 30, 42, 56, 72...

On the EDPT, recognizing these numbers instantly can save you from performing tedious second-difference calculations. If you see the sequence 3, 7, 13, 21, 31, 43..., notice how each number is exactly 1 greater than a pronic number ($n(n+1) + 1$, or $n^2 + n + 1$). The next term is $6(7) + 1 = 43$, followed by $7(8) + 1 = 57$.

Multi-Step and Progressive Operator Sequences

Another challenging variety of nonlinear sequences involves alternating or progressive mathematical operators. In these patterns, the action performed on each term changes according to a secondary sequence.

Alternating Operators

Consider the sequence: 3, 4, 8, 9, 18, 19, 38, ...

• Here, the rules alternate between adding and multiplying:
  • 3 + 1 = 4
  • 4 * 2 = 8
  • 8 + 1 = 9
  • 9 * 2 = 18
  • 18 + 1 = 19
  • 19 * 2 = 38
• The pattern of operations is (+1, *2, +1, *2...). The next term is $38 + 1 = 39$.

Progressive Operators

Consider the sequence: 2, 3, 6, 18, 108, ...

• In this sequence, each number is multiplied by the preceding number to generate the next:
  • 2 * 3 = 6
  • 3 * 6 = 18
  • 6 * 18 = 108
• This is a multiplicative version of the Fibonacci sequence. The next term is $18 * 108 = 1944$.

Consider another progression: 1, 2, 6, 21, 88, ...

• The operators are increasing arithmetically:
  • (1 * 1) + 1 = 2
  • (2 * 2) + 2 = 6
  • (6 * 3) + 3 = 21
  • (21 * 4) + 4 = 88
• The next term is $(88 * 5) + 5 = 440 + 5 = 445$.

Factorials and Exponential Scale Growth

When the terms of a sequence escalate at an extremely rapid rate, arithmetic and standard geometric operations should be ruled out in favor of factorial or exponential growth.

• Factorial sequence ($n!$): 1, 2, 6, 24, 120, 720, 5040...
  • Spurred by multiplying the previous term by $n$: $1 \times 2 = 2$; $2 \times 3 = 6$; $6 \times 4 = 24$; $24 \times 5 = 120$.
• Variable-base power sequence ($n^n$): 1, 4, 27, 256, 3125...
  • $1^1, 2^2, 3^3, 4^4, 5^5$. On the EDPT, if you observe the fifth or sixth term surpassing 1,000 when starting from a single digit, check for factorial products or $n^n$ bases immediately.

The Method of Second Differences

When a sequence makes absolutely no sense—it's not arithmetic, not geometric, not alternating, and not a simple square/cube—your ultimate fallback weapon is the Method of Second Differences.

Sometimes, the sequence of differences between the numbers forms its own arithmetic sequence.

How to Calculate Second Differences

Consider the sequence: 2, 4, 8, 14, 22, ...

1. Find the First Differences:

   • 4 - 2 = 2
   • 8 - 4 = 4
   • 14 - 8 = 6
   • 22 - 14 = 8

   The sequence of first differences is: 2, 4, 6, 8...

2. Analyze the First Differences:

   • We can see this is an arithmetic progression (+2).
   • Therefore, the next first difference should be 10.

3. Apply the Next Difference to the Original Sequence:

   • Original last term (22) + next difference (10) = 32.

Let's try a harder one: 1, 6, 15, 28, 45, ...

1. First Differences:

   • 6 - 1 = 5
   • 15 - 6 = 9
   • 28 - 15 = 13
   • 45 - 28 = 17
   • First differences: 5, 9, 13, 17.

2. Second Differences (Difference of the differences):

   • 9 - 5 = 4
   • 13 - 9 = 4
   • 17 - 13 = 4
   • The second difference is a constant +4.

3. Calculate Upward:

   • The next first difference is $17 + 4 = 21$.
   • The next term in the main sequence is $45 + 21 = 66$.

If you find a constant second difference, the original sequence is governed by a quadratic equation (like $an^2 + bn + c$). You do not need to find the equation; you just need to calculate the differences to find the next number.

The 15-Second Diagnostic Flowchart

Under the strict time limits of the EDPT, you cannot afford to waste minutes guessing rules. Use this rapid diagnostic sequence for every number pattern:

1. Check Growth Rate: Does the sequence increase slowly (addition), moderately (multiplication/quadratic), or explosively (exponents/factorials)?
2. Scan for Squares/Cubes: Are the numbers within $\pm 2$ of $1, 4, 9, 16, 25...$ or $1, 8, 27, 64...$?
3. Map First Differences: Spend 5 seconds calculating the differences between the first 4 terms. Is the sequence of differences simple (+2, +4, +6)?
4. Map Second Differences: If the first differences form a pattern but not a constant one, check their differences. A constant second difference guarantees a quadratic sequence.
5. Check Adjacent Addition: Try adding terms 1 and 2 to see if they equal term 3, and terms 2 and 3 to see if they equal term 4.
6. Look for Alternating Operations: If the numbers rise, fall, and rise again, look for alternating (+ then -) or alternating indices.

Strategic Tips for Nonlinear Sequences

1. Memorize the Milestones: If you see numbers near 25, 36, 49, 64—immediately think about squares. If you see numbers near 27, 64, 125—think about cubes.
2. The Growth Rate Check: If a sequence grows faster than addition (not arithmetic) but slower than pure multiplication (not geometric), it is highly likely to be a polynomial sequence solvable via second differences.
3. Don't Panic on Strange Numbers: If the sequence starts with seemingly random leaps, immediately map out the first differences. The underlying pattern usually reveals itself on the second tier.