**GMAT Focus Edition: Algebra – Functions & Sequences**

GMAT Focus Edition: Algebra – Functions & Sequences

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What This Is

Functions and sequences appear in ~5–7% of GMAT Focus Quantitative questions, often disguised as word problems, Data Sufficiency, or abstract algebra. Mastering them boosts your efficiency (you’ll solve faster) and accuracy (you’ll avoid traps). A typical question:

For every positive integer n, the function f is defined by f(n) = n² – 3n + 2. What is the value of f(5) – f(3)?

Why it matters: These questions test pattern recognition, substitution, and algebraic manipulation—skills that separate 600 scorers from 700+ scorers.

Key Concepts & Techniques

1. Function Notation (f(x))
2. A rule that assigns an input to exactly one output.

3. When to use: Any question with f(x), g(x), or h(x). Substitute the input value directly into the rule.

4. Sequence Definition (aₙ)

5. A list of numbers where each term depends on its position (n).

6. When to use: Questions asking for the nth term, sum of terms, or pattern in a sequence.

7. Recursive Sequences

8. Defined by a starting term and a rule for the next term (e.g., aₙ₊₁ = aₙ + 2).

9. When to use: When the sequence is given in terms of previous terms (e.g., "each term is 3 more than the previous term").

10. Arithmetic Sequences

11. Each term increases by a constant difference (d).
12. Formula: aₙ = a₁ + (n – 1)d

13. When to use: Questions about evenly spaced numbers (e.g., salaries increasing by $500/year).

14. Geometric Sequences

15. Each term is multiplied by a constant ratio (r).
16. Formula: aₙ = a₁ × rⁿ⁻¹

17. When to use: Questions about exponential growth (e.g., investments doubling every year).

18. Sum of Sequences

19. Arithmetic sum: Sₙ = n/2 × (a₁ + aₙ)
20. Geometric sum (finite): Sₙ = a₁(1 – rⁿ)/(1 – r)

21. When to use: Questions asking for the total of a series (e.g., "sum of the first 10 terms").

22. Function Composition (f(g(x)))

23. Plugging one function into another.

24. When to use: Questions like "If f(x) = 2x + 1 and g(x) = x², what is f(g(3))?"

25. Piecewise Functions

26. Different rules for different input ranges (e.g., f(x) = x² if x < 0, f(x) = 2x if x ≥ 0).
27. When to use: Questions with "if" conditions in the function definition.

Step-by-Step Strategy

Every time you see a function or sequence question, follow these steps:

1. Identify the Type
2. Is it a function (f(x)) or a sequence (aₙ)?

3. If a sequence, is it arithmetic, geometric, or recursive?

4. Write Down the Given Rule

5. For functions: f(x) = [rule].

6. For sequences: aₙ = [rule] or aₙ₊₁ = [rule].

7. Substitute or Expand

8. For functions: Plug in the input value (e.g., f(5) = 5² – 3(5) + 2).

9. For sequences: Find the required term (e.g., a₅ = a₁ + 4d).

10. Simplify or Solve

11. Perform arithmetic or algebraic manipulation to get the answer.

12. For sums, use the appropriate formula.

13. Check for Traps

14. Did you misapply the rule? (e.g., confusing f(x) with f(n)).
15. Did you miscount terms? (e.g., "first 5 terms" vs. "5th term").

Fully Worked Example

Question:
For every positive integer n, the function f is defined by f(n) = n² – 3n + 2. What is the value of f(5) – f(3)?

Solution (using the strategy):

1. Identify the Type: This is a function (f(n)).
2. Write Down the Rule: f(n) = n² – 3n + 2.
3. Substitute:
4. f(5) = 5² – 3(5) + 2 = 25 – 15 + 2 = 12
5. f(3) = 3² – 3(3) + 2 = 9 – 9 + 2 = 2
6. Simplify: f(5) – f(3) = 12 – 2 = 10
7. Check for Traps: Did you misapply the rule? No. Did you miscalculate? No.

Answer: 10

Common Mistakes

1. Mistake: Confusing f(x) with f(n).
2. Why it happens: Students see "n" and assume it’s a sequence, not a function.

3. Correct approach: f(n) is still a function—just with n as the input.

4. Mistake: Misapplying sequence formulas.

5. Why it happens: Using the arithmetic sum formula for a geometric sequence (or vice versa).

6. Correct approach: Always confirm the sequence type first.

7. Mistake: Off-by-one errors in sequences.

8. Why it happens: Counting the first term as "term 0" instead of "term 1."

9. Correct approach: The first term is a₁, not a₀.

10. Mistake: Ignoring piecewise conditions.

11. Why it happens: Forgetting to check which rule applies (e.g., x < 0 vs. x ≥ 0).
12. Correct approach: Highlight the conditions before substituting.

GMAT Traps & Timing

1. Trap: Hidden Recursive Definitions
2. The test may define a sequence recursively (e.g., "each term is 2 more than the previous term") but ask for a term far out (e.g., the 100th term).

3. Avoid it: Recognize when to use the explicit formula (aₙ = a₁ + (n – 1)d) instead of calculating all terms.

4. Trap: Function Composition Misordering

5. f(g(x)) is not the same as g(f(x)).

6. Avoid it: Always work from the inside out (e.g., g(x) first, then f).

7. Timing:

8. Functions: 1–1.5 minutes per question.
9. Sequences: 1.5–2 minutes (longer if sums are involved).

Quick Practice

1. Question:
   The sequence aₙ is defined by a₁ = 3 and aₙ₊₁ = 2aₙ for n ≥ 1. What is the value of a₄?
2. (A) 6
3. (B) 12
4. (C) 24
5. (D) 48
6. (E) 96

Answer: (C) 24
Explanation: This is a geometric sequence with a₁ = 3 and r = 2. a₄ = 3 × 2³ = 24.

1. Question:
   If f(x) = 2x + 1 and g(x) = x² – 1, what is f(g(2))?
2. (A) 5
3. (B) 7
4. (C) 9
5. (D) 11
6. (E) 13

Answer: (B) 7
Explanation: g(2) = 2² – 1 = 3. Then f(3) = 2(3) + 1 = 7.

Last-Minute Cram Sheet

1. Function substitution: f(x) = [rule] → plug in x.
2. Arithmetic sequence: aₙ = a₁ + (n – 1)d.
3. Geometric sequence: aₙ = a₁ × rⁿ⁻¹.
4. Arithmetic sum: Sₙ = n/2 × (a₁ + aₙ).
5. Geometric sum (finite): Sₙ = a₁(1 – rⁿ)/(1 – r).
6. Recursive sequences: Write out terms if n is small; use explicit formula if n is large.
7. f(g(x)) ≠ g(f(x)) — order matters!
8. Piecewise functions: Check the condition before substituting.
9. Off-by-one errors: First term is a₁, not a₀.
10. ⚠️ Trap: If a sequence is defined recursively, don’t assume it’s arithmetic or geometric—verify!

Final Tip: On test day, write down the rule before substituting. This prevents careless errors and keeps you organized.