**GRE Advanced Topics: Number Line & Sequences – Arithmetic/Geometric Sequences & Consecutive Integers**

GRE Advanced Topics: Number Line & Sequences – Arithmetic/Geometric Sequences & Consecutive Integers

By an Elite GRE Instructor (320+ Scorer Guarantee)

What This Is

The GRE tests number line reasoning and sequences (arithmetic, geometric, consecutive integers) to assess your ability to manipulate patterns, inequalities, and algebraic expressions under time pressure. These questions appear in Quantitative Comparison (QC), Problem Solving (PS), and Data Interpretation (DI) formats, often disguised as word problems or abstract algebra. Mastering them can boost your Quant score by 2–4 points by eliminating careless errors and exploiting ETS’s predictable traps.

Example GRE-Style Question:
If the sum of 5 consecutive integers is 0, what is the greatest of these integers? (A) –2 (B) –1 (C) 0 (D) 1 (E) 2

(Answer: E. The integers are –2, –1, 0, 1, 2; the greatest is 2.)

Key Concepts & Techniques

1. Arithmetic Sequence Formula
2. What it is: A sequence where each term increases by a constant difference (d).
   Formula: aₙ = a₁ + (n–1)d
3. When to use: When given a sequence with a common difference (e.g., 3, 7, 11, 15…) or asked for the nth term/sum.

4. Pro tip: For sums, use Sₙ = n/2 (a₁ + aₙ) or Sₙ = n/2 [2a₁ + (n–1)d].

5. Geometric Sequence Formula

6. What it is: A sequence where each term is multiplied by a constant ratio (r).
   Formula: aₙ = a₁ × rⁿ⁻¹
7. When to use: When terms grow exponentially (e.g., 2, 6, 18, 54…) or involve percentages/fractions.

8. Pro tip: Sum of first n terms: Sₙ = a₁(1–rⁿ)/(1–r) (if r ≠ 1).

9. Consecutive Integers: Representation

10. What it is: Integers that follow each other without gaps (e.g., x, x+1, x+2).
11. When to use: When the problem mentions "consecutive integers," "even/odd consecutive numbers," or "sum/product of consecutive terms."

12. Pro tip: For k consecutive integers, the sum is always divisible by k if k is odd.

13. Number Line Inequalities

14. What it is: Visualizing ranges (e.g., –3 < x ≤ 5) or absolute value (|x–2| < 4).
15. When to use: For QC questions comparing expressions or PS questions with ranges.

16. Pro tip: Always test boundary values (e.g., x = –3 and x = 5 for –3 ≤ x ≤ 5).

17. Sum of Consecutive Integers Shortcut

18. What it is: For n consecutive integers starting at a, sum = n × (first + last)/2.
19. When to use: When asked for the sum of a sequence (e.g., "sum of 10 consecutive integers is 100").

20. Pro tip: If the sum is 0, the sequence is symmetric around 0 (e.g., –2, –1, 0, 1, 2).

21. Even/Odd Consecutive Integers

22. What it is: Even: 2x, 2x+2, 2x+4; Odd: 2x+1, 2x+3, 2x+5.
23. When to use: When the problem specifies "consecutive even/odd integers."

24. Pro tip: The sum of k consecutive odd integers is k² if k is odd (e.g., 1+3+5=9=3²).

25. Plugging in Numbers for Sequences

26. What it is: Testing small values (e.g., n=1, n=2) to identify patterns.
27. When to use: For QC questions or when the sequence is defined recursively (e.g., aₙ = aₙ₋₁ + 3).
28. Pro tip: Avoid n=0 unless the problem allows it (GRE sequences often start at n=1).

Step-by-Step Strategy

Follow these steps for every sequence/number line question:

1. Identify the Sequence Type
2. Is it arithmetic (common difference d), geometric (common ratio r), or consecutive integers?

3. Example: "The first term is 2, and each term increases by 3" → Arithmetic (d=3).

4. Write the General Formula

5. Arithmetic: aₙ = a₁ + (n–1)d
6. Geometric: aₙ = a₁ × rⁿ⁻¹

7. Consecutive integers: x, x+1, x+2, … or 2x, 2x+2, … (even/odd).

8. Translate the Problem into an Equation

9. Sum of terms? → Use sum formula.
10. Specific term? → Plug into aₙ.

11. Range/inequality? → Draw a number line.

12. Solve for the Unknown

13. Isolate variables; check for extraneous solutions (e.g., r=1 in geometric sums).

14. For QC, compare both sides algebraically or plug in values.

15. Verify with Boundary Cases

16. Test n=1, n=2 or edge values (e.g., x=0 for inequalities).

17. Example: If |x–3| < 5, test x=–2 and x=8 to confirm the range.

18. Eliminate Traps

19. Watch for:
    • Off-by-one errors (e.g., n vs. n–1 in formulas).
    • Negative ratios (r < 0) in geometric sequences.
    • Non-integer solutions (GRE answers are usually integers).

Fully Worked GRE-Style Example

Question:
In an arithmetic sequence, the 4th term is 14 and the 10th term is 38. What is the sum of the first 20 terms?

Step 1: Identify the Sequence Type
- Arithmetic sequence (common difference d).

Step 2: Write the General Formula
- aₙ = a₁ + (n–1)d

Step 3: Translate into Equations
- 4th term: a₄ = a₁ + 3d = 14 - 10th term: a₁₀ = a₁ + 9d = 38

Step 4: Solve for a₁ and d
- Subtract the first equation from the second: (a₁ + 9d) – (a₁ + 3d) = 38 – 14 → 6d = 24 → d = 4 - Plug d=4 into a₄: a₁ + 3(4) = 14 → a₁ = 2

Step 5: Find the 20th Term
- a₂₀ = a₁ + 19d = 2 + 19(4) = 78

Step 6: Calculate the Sum
- Sum formula: S₂₀ = 20/2 (a₁ + a₂₀) = 10(2 + 78) = 800

Answer: 800

Common Mistakes

1. Mistake: Using the wrong formula for the sum of an arithmetic sequence.
2. Why it happens: Confusing Sₙ = n/2 (a₁ + aₙ) with Sₙ = n/2 [2a₁ + (n–1)d].

3. Correct approach: Memorize both; use the first if you know aₙ, the second if you know d.

4. Mistake: Misrepresenting consecutive integers (e.g., writing x, x+1, x+3 for consecutive odds).

5. Why it happens: Forgetting that consecutive odds/evens skip a number.

6. Correct approach: Odds: 2x+1, 2x+3, 2x+5; Evens: 2x, 2x+2, 2x+4.

7. Mistake: Ignoring negative ratios in geometric sequences.

8. Why it happens: Assuming r is positive (e.g., r=–2 flips signs every term).

9. Correct approach: Test r > 0 and r < 0 if the problem allows.

10. Mistake: Off-by-one errors in term numbering.

11. Why it happens: Counting the first term as a₀ instead of a₁.

12. Correct approach: GRE sequences almost always start at n=1.

13. Mistake: Forgetting to check boundary values in inequalities.

14. Why it happens: Solving |x–2| < 3 as –3 < x–2 < 3 but not testing x=–1 and x=5.
15. Correct approach: Always plug in the endpoints to confirm the range.

GRE Traps & Timing

1. Trap: "Consecutive" vs. "Consecutive Even/Odd"
2. Example: "The sum of 3 consecutive integers is 30" vs. "The sum of 3 consecutive even integers is 30."

3. Avoid it: Represent evens/odds as 2x, 2x+2, 2x+4 (not x, x+2, x+4).

4. Trap: Geometric Sequence with r=1

5. Example: "The sum of the first 5 terms of a geometric sequence is 10. If the first term is 2, what is the common ratio?"

6. Avoid it: If r=1, the sum is 5×2=10 (valid solution). Don’t assume r≠1.

7. Trap: Number Line Inequalities with Absolute Value

8. Example: |x–1| > 3 → x > 4 or x < –2 (not –2 < x < 4).

9. Avoid it: Split into two cases: x–1 > 3 and x–1 < –3.

10. Timing:

11. QC: 60–90 seconds (plug in numbers if stuck).
12. PS: 90–120 seconds (write formulas first).
13. DI: 120 seconds (scan for sequence patterns in tables/graphs).

Quick Practice

1. The sum of 4 consecutive odd integers is 48. What is the smallest integer?
2. (A) 7 (B) 9 (C) 11 (D) 13 (E) 15

3. Answer: B (9, 11, 13, 15; sum = 48).

4. In a geometric sequence, the 3rd term is 12 and the 6th term is 96. What is the 1st term?

5. (A) 2 (B) 3 (C) 4 (D) 6 (E) 8
6. Answer: B (a₃ = a₁r² = 12; a₆ = a₁r⁵ = 96; divide to find r=2, then a₁=3).

Last-Minute Cram Sheet

1. Arithmetic sequence: aₙ = a₁ + (n–1)d; Sum = n/2 (a₁ + aₙ).
2. Geometric sequence: aₙ = a₁ × rⁿ⁻¹; Sum = a₁(1–rⁿ)/(1–r) (r≠1).
3. Consecutive integers: x, x+1, x+2; Sum of k terms = k × (first + last)/2.
4. Consecutive evens/odds: 2x, 2x+2, 2x+4 or 2x+1, 2x+3, 2x+5.
5. Sum of k consecutive odds (if k odd): k² (e.g., 1+3+5=9=3²).
6. Absolute value inequality: |x–a| < b → a–b < x < a+b; |x–a| > b → x < a–b or x > a+b.
7. GRE sequences start at n=1 (not n=0).
8. If the sum of consecutive integers is 0, the sequence is symmetric around 0.
9. For QC, plug in n=1, n=2 to test patterns.
10. ⚠️ Trap: "Consecutive" ≠ "consecutive even/odd" — represent them differently!