Section 3.1: Arithmetic and Geometric Progressions

Arithmetic and Geometric Progressions

When preparing for the Electronic Data Processing Test (EDPT), mastering number sequences is absolutely essential. Number sequences are designed to test your logical reasoning, pattern recognition, and mathematical agility—all core skills for computer programming and IT roles. The most fundamental types of sequences you will encounter are arithmetic and geometric progressions. While they may seem straightforward, the extreme time constraints of the EDPT require you to recognize and solve them almost instinctively.

Understanding Arithmetic Progressions

An arithmetic progression (or arithmetic sequence) is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant is known as the common difference ($d$). In simpler terms, you are either adding or subtracting the exact same number each time to get to the next term.

The Anatomy of an Arithmetic Sequence

Consider the sequence: 5, 9, 13, 17, 21, ...

To identify the pattern, look at the difference between consecutive terms:

• 9 - 5 = 4
• 13 - 9 = 4
• 17 - 13 = 4
• 21 - 17 = 4

Since the difference is a constant +4, this is an arithmetic progression with a common difference of 4. The next term would simply be $21 + 4 = 25$.

Decreasing Arithmetic Progressions

Arithmetic sequences can also decrease. Consider the sequence: 82, 75, 68, 61, 54, ...

Here, the operation is subtraction:

• 75 - 82 = -7
• 68 - 75 = -7
• 61 - 68 = -7

The common difference here is -7. The next term would be $54 - 7 = 47$.

Strategic Tips for Arithmetic Progressions on the EDPT

1. First Differences First: The very first step you should take when seeing any number sequence on the EDPT is to mentally calculate the difference between the first two terms, and then the second and third terms. If they match, you almost certainly have an arithmetic progression.
2. Don't Stop at Two: A common EDPT trap is a sequence like 2, 4, 8... where the difference between 2 and 4 is 2, but the difference between 4 and 8 is 4. If you only check the first two numbers and assume it's $+2$, you will fail the question. Always verify the pattern with at least three terms.
3. Negative Numbers: Be prepared for sequences that cross the zero threshold (e.g., 5, 2, -1, -4, -7). Do not let the presence of negative numbers slow your calculations.

Understanding Geometric Progressions

A geometric progression (or geometric sequence) is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio ($r$).

The Anatomy of a Geometric Sequence

Consider the sequence: 3, 6, 12, 24, 48, ...

To identify the pattern, look at the ratio between consecutive terms:

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

Since the ratio is a constant 2, this is a geometric progression with a common ratio of 2. The next term would be $48 * 2 = 96$.

Fractional Geometric Progressions (Division)

Geometric sequences can also involve division, which is the same as multiplying by a fraction. Consider the sequence: 400, 100, 25, 6.25, ...

Here, the operation is division by 4 (or multiplication by 0.25):

• 100 / 400 = 1/4
• 25 / 100 = 1/4
• 6.25 / 25 = 1/4

The next term would be $6.25 / 4 = 1.5625$.

Alternating Signs in Geometric Sequences

A classic EDPT curveball is a geometric sequence with a negative common ratio, which causes the signs to alternate. Consider: 2, -6, 18, -54, 162, ...

The common ratio here is -3. If you see signs alternating (+, -, +, -, +), immediately suspect a geometric sequence with a negative ratio.

Comparison Table: Arithmetic vs. Geometric

Recognizing the Difference Quickly

On the EDPT, speed is everything. You do not have time to write out formulas. You must develop a "feel" for the numbers.

• Slow, steady growth or decline strongly suggests an arithmetic progression. If a sequence starts at 12 and five steps later is only at 42, it's likely arithmetic.
• Explosive growth or rapid shrinking strongly suggests a geometric progression. If a sequence starts at 3 and five steps later is at 729, it's definitely geometric.
• Always check the first three terms. This cannot be stated enough. The sequence 3, 6... could be arithmetic (+3) leading to 9, or it could be geometric (*2) leading to 12. Only the third term resolves the ambiguity.

Worked Examples

Example 1: Find the next number in the sequence: 14, 23, 32, 41, 50, ?

1. Check first differences: $23 - 14 = 9$.
2. Verify with the next pair: $32 - 23 = 9$.
3. Verify again: $41 - 32 = 9$.
4. It is an arithmetic progression with $d = 9$.
5. The next term is $50 + 9 = 59$.

Example 2: Find the next number in the sequence: 5, -10, 20, -40, 80, ?

1. Notice the alternating signs immediately. This implies multiplication by a negative number.
2. Check the ratio: $-10 / 5 = -2$.
3. Verify: $20 / -10 = -2$.
4. Verify: $-40 / 20 = -2$.
5. It is a geometric progression with $r = -2$.
6. The next term is $80 * -2 = -160$.

By mastering these foundational patterns, you build the cognitive framework necessary to tackle the much more complex multi-step and nonlinear sequences that follow in the next sections.