Study guide
Section 3.1: Arithmetic and Geometric Progressions
Electronic Data Processing Test EDPT
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1 Chapter 1: Verbal Reasoning
1.1 Word Analogies 1.2 Reading Comprehension and Deductions 1.3 Verbal Classification and Associations
2 Chapter 2: Arithmetic Reasoning
Section 2.1: Algebraic Word Problems Section 2.2: Ratios, Proportions, and Rates Section 2.3: Percentages, Fractions, and Decimals Section 2.4: Divisibility and Prime Numbers in Word Problems
3 Chapter 3: Number Sequences
Section 3.1: Arithmetic and Geometric Progressions Section 3.2: Multi-Step and Alternating Sequences Section 3.3: Nonlinear Sequences Section 3.4: Grid-Based and Matrix Sequences
4 Chapter 4: Figural Reasoning
4.1 Spatial Rotations and Reflections 4.2 Figural Progressions and Series 4.3 Figural Analogies and Matrices
5 Chapter 5: Logical Analysis
5.1 Conditional Logic and Truth Tables 5.2 Syllogisms and Venn Diagrams
Section 3.1: Arithmetic and Geometric Progressions
Key Takeaways Explain This Topic AI
Arithmetic progressions involve adding or subtracting a constant value (common difference) between consecutive terms.
Geometric progressions involve multiplying or dividing by a constant value (common ratio) between consecutive terms.
Quickly identifying a rule that governs a simple progression is a useful symbolic-reasoning skill.
Always calculate the difference or ratio between the first three terms to confirm the pattern before extending it.
A common trap is assuming an arithmetic sequence when a geometric one is at play, especially with small initial numbers.
Last updated: July 2026
Arithmetic and Geometric Progressions
Number-sequence drills are one practical way this guide develops symbolic reasoning for EDPT preparation. 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, timed practice rewards fluent recognition and solution methods.
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 d 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 21 + 4 = 25 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 54 - 7 = 47 54 − 7 = 47 .
Strategic Tips for Arithmetic-Progression Practice
First Differences First: The very first step you should take when seeing a number sequence in practice 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.
Don't Stop at Two: A common sequence-practice trap is 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 +2 + 2 , you will fail the question. Always verify the pattern with at least three terms.
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 r 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 48 * 2 = 96 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 6.25 / 4 = 1.5625 6.25/4 = 1.5625 .
Alternating Signs in Geometric Sequences
A classic sequence variation 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
- Feature Arithmetic Progression Geometric Progression
- Core Operation Addition or Subtraction Multiplication or Division
- Constant Term Common Difference ( d d d ) Common Ratio ( r r r )
- Growth Rate Linear (Steady, straight line) Exponential (Rapidly steepening curve)
- Typical Practice Trap Confusing early terms with geometric (e.g., 2, 4...) Alternating signs (negative ratio)
Formula for Nth Term a n = a 1 + ( n − 1 ) d a_n = a_1 + (n-1)d a n = a 1 + ( n − 1 ) d a n = a 1 ∗ r ( n − 1 ) a_n = a_1 * r^{(n-1)} a n = a 1 ∗ r ( n − 1 )
Recognizing the Difference Quickly
In timed practice, fluency matters. 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, ?
Check first differences: 23 − 14 = 9 23 - 14 = 9 23 − 14 = 9 .
Verify with the next pair: 32 − 23 = 9 32 - 23 = 9 32 − 23 = 9 .
Verify again: 41 − 32 = 9 41 - 32 = 9 41 − 32 = 9 .
It is an arithmetic progression with d = 9 d = 9 d = 9 .
The next term is 50 + 9 = 59 50 + 9 = 59 50 + 9 = 59 .
Example 2: Find the next number in the sequence: 5, -10, 20, -40, 80, ?
Notice the alternating signs immediately. This implies multiplication by a negative number.
Check the ratio: − 10 / 5 = − 2 -10 / 5 = -2 − 10/5 = − 2 .
Verify: 20 / − 10 = − 2 20 / -10 = -2 20/ − 10 = − 2 .
Verify: − 40 / 20 = − 2 -40 / 20 = -2 − 40/20 = − 2 .
It is a geometric progression with r = − 2 r = -2 r = − 2 .
The next term is 80 ∗ − 2 = − 160 80 * -2 = -160 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.