Section 3.2: Multi-Step and Alternating Sequences

Multi-Step and Alternating Sequences

Once you are comfortable with basic arithmetic and geometric progressions, the EDPT will introduce complexity by combining them. Single-operation sequences are relatively rare in the harder sections of the exam. Instead, you will encounter multi-step sequences and alternating sequences. Recognizing these structures quickly is critical for maintaining your pace on the test.

Multi-Step Sequences

A multi-step sequence is a single progression of numbers governed by a repeating cycle of different mathematical operations. Instead of a single rule like "add 4," the rule might be "add 2, then multiply by 3, then subtract 1, and repeat."

Analyzing Multi-Step Patterns

Consider this sequence: 4, 6, 18, 20, 60, 62, ...

If we try to find a single common difference or ratio, we fail:

• Differences: +2, +12, +2, +40, +2
• Ratios: *1.5, *3, *1.11, *3, *1.03

Neither a single arithmetic nor a single geometric rule applies. However, look closely at the differences and ratios combined:

• 4 to 6 is +2
• 6 to 18 is x3
• 18 to 20 is +2
• 20 to 60 is x3
• 60 to 62 is +2

The pattern is a repeating cycle: [+2, x3]. Since the last operation used was +2 (going from 60 to 62), the next operation must be x3. Therefore, the next term is $62 * 3 = 186$.

Strategic Tips for Multi-Step Sequences

1. Look for Pairs: Multi-step sequences often involve pairs of numbers that relate easily to each other. In the example above, 4 and 6 relate, 18 and 20 relate, 60 and 62 relate. Recognizing these "clusters" can help you spot the interim operations (like the x3 jump between the pairs).
2. Map the Operations: Literally write (or mentally map) the operation above each transition. If you see +3, -1, +3, -1, you've cracked the code.
3. Track Your Position: The most common mistake in multi-step sequences is applying the wrong part of the rule to find the final answer. If the rule is [+4, /2], and the sequence ends after a /2 step, make sure you apply +4 to find the next number.

Alternating (Interwoven) Sequences

An alternating sequence is actually two distinct sequences woven together into a single line. This is one of the most common and confusing sequence types on the EDPT.

The Anatomy of an Alternating Sequence

Consider this sequence: 2, 50, 4, 45, 8, 40, 16, 35, ...

If you try to analyze this as a single sequence, it looks like pure chaos: up 48, down 46, up 41, down 37, etc. There is no mathematical sense here.

However, when a sequence jumps wildly up and down, your immediate instinct should be to check for an alternating sequence. To do this, "unweave" the sequence by looking at every other number.

Let's extract the odd-positioned terms (1st, 3rd, 5th, 7th):

• 2, 50, 4, 45, 8, 40, 16, 35
• Sub-sequence A: 2, 4, 8, 16 (Rule: Multiply by 2)

Now let's extract the even-positioned terms (2nd, 4th, 6th, 8th):

• 2, 50, 4, 45, 8, 40, 16, 35
• Sub-sequence B: 50, 45, 40, 35 (Rule: Subtract 5)

To find the next term in the overall sequence (the 9th term), we look at Sub-sequence A. The next number should be $16 * 2 = 32$. If we needed the 10th term, we would look at Sub-sequence B: $35 - 5 = 30$.

Tricky Variations of Alternating Sequences

Sometimes, the two interwoven sequences are related to each other, or they might be three interwoven sequences (though rare on the EDPT due to time constraints).

Example: The "Same Rule" Alternating Sequence 5, 10, 8, 13, 11, 16, ...

Unweaving:

• Sub A (Odds): 5, 8, 11 (Rule: +3)
• Sub B (Evens): 10, 13, 16 (Rule: +3)

Both sub-sequences follow the same rule (+3), just starting from different numbers. The next term (7th position) belongs to Sub A: $11 + 3 = 14$.

(Note: This specific sequence can also be solved as a multi-step sequence: +5, -2, +5, -2, +5... Both methods yield 14. Use whichever pattern your brain recognizes first!)

How to Spot Alternating Sequences Quickly

1. The See-Saw Effect: The numbers bounce up and down dramatically (e.g., small, huge, medium-small, slightly-less-huge). Standard progressions go in one direction.
2. The Length: Alternating sequences require more numbers to establish a pattern. If a question presents you with 7 or 8 numbers instead of the usual 4 or 5, it is very likely an alternating sequence.
3. Unrelated Number Types: If a sequence mixes fractions and whole numbers, or negative numbers and positive numbers in a non-geometric way, try unweaving them.

Worked Examples

Example 1: Find the next number: 10, 5, 8, 4, 7, 3.5, ?

1. Analyze transitions: 10 to 5 (/2), 5 to 8 (+3), 8 to 4 (/2), 4 to 7 (+3), 7 to 3.5 (/2).
2. The pattern is a multi-step cycle: [/2, +3].
3. The last operation was /2, so the next operation must be +3.
4. The next term is $3.5 + 3 = 6.5$.

Example 2: Find the next number: 1, 99, 3, 90, 9, 81, 27, ?

1. The sequence jumps up and down. Let's unweave it.
2. Sub A (Odds): 1, 3, 9, 27 (Rule: *3)
3. Sub B (Evens): 99, 90, 81 (Rule: -9)
4. The missing term is in the 8th position (an even position), so it belongs to Sub B.
5. The next term is $81 - 9 = 72$.

By practicing the identification of multi-step and alternating patterns, you will develop the cognitive flexibility required to score highly on the EDPT's logic sections.