GMAT Algebra: Functions, Sequences
Last updated: May 2, 2026
Algebra: Functions, Sequences questions are one of the highest-leverage areas to study for the GMAT. This guide breaks down the rule, the elements you need to recognize, the named traps that catch most students, and a memory aid that scales to test day. Read it once, then practice the same sub-topic adaptively in the app.
The rule
A function is a rule that turns each input into exactly one output, and a sequence is just a function whose inputs are positive integers (the term numbers). Your job on the GMAT is to read the rule precisely, identify what the input actually is, and then substitute carefully — most errors here are mechanical, not conceptual. For sequences, decide first whether the rule is explicit (gives $a_n$ directly from $n$) or recursive (gives $a_n$ from earlier terms), because that choice dictates whether you can jump straight to a term or must build up.
Elements breakdown
Function evaluation
Substitute the given input wherever the variable appears in the rule, then simplify.
- Replace every instance of the variable
- Wrap substituted expressions in parentheses
- Apply order of operations carefully
- Simplify exponents before multiplication
- Track signs through each step
Common examples:
- If $f(x) = x^2 - 3x$, then $f(-2) = (-2)^2 - 3(-2) = 4 + 6 = 10$.
Composite functions
$f(g(x))$ means evaluate $g$ first, then feed that output into $f$.
- Evaluate the inner function first
- Treat the inner output as a single input
- Substitute that output into the outer rule
- Do NOT multiply $f$ by $g$
- Watch for $f(g(x)) \ne g(f(x))$
Explicit sequence formulas
A rule that gives $a_n$ directly as a function of $n$.
- Identify the formula form
- Substitute the term number for $n$
- Compute without building earlier terms
- Use for arithmetic: $a_n = a_1 + (n-1)d$
- Use for geometric: $a_n = a_1 \cdot r^{n-1}$
Recursive sequence formulas
A rule that defines $a_n$ in terms of one or more earlier terms.
- Note the starting term(s)
- Apply the recurrence step by step
- Build a small table of terms
- Look for a repeating cycle
- Do NOT skip ahead until you spot a pattern
Domain and range awareness
Check what inputs are legal and what outputs are possible.
- Reject inputs that produce division by zero
- Reject inputs that produce negative under even roots
- Note when outputs are bounded
- Verify the input is in the stated domain
- Match output type to the answer choices
Common patterns and traps
The Compound Input Trap
The function is defined simply, like $f(x) = x^2 + 1$, but you are asked for $f(a + b)$ or $f(2x - 3)$. Students who substitute without parentheses produce $a + b^2 + 1$ instead of $(a+b)^2 + 1 = a^2 + 2ab + b^2 + 1$. The trap exploits the fact that exponents bind more tightly than addition.
A wrong choice that drops the cross-term $2ab$ when squaring a binomial input, or that distributes an exponent across a sum.
The Recursive Slow Burn
You are given $a_1$ and $a_n = a_{n-1} + k$ (or a similar recurrence) and asked for $a_{40}$ or the sum of the first 30 terms. The intended approach is to recognize the arithmetic or geometric structure and use a closed form; the trap is to start writing out terms and get pulled into a five-minute computation. The GMAT explicitly rewards pattern recognition here.
A wrong choice that matches what you would get after computing only 4 or 5 terms of a much longer sequence — i.e., the answer if you stopped early.
Composition Order Reversal
In $f(g(x))$, the inner function $g$ runs first. Students who read left-to-right sometimes evaluate $f$ first instead, producing $g(f(x))$. Because $f \circ g$ and $g \circ f$ usually give different outputs, this generates a tempting wrong answer that the test writers will list among the choices.
A wrong choice that equals $g(f(x))$ for the given input — same numbers, just composed in the wrong order.
The Off-By-One on Term Index
Arithmetic sequence formulas use $a_n = a_1 + (n-1)d$, with $(n-1)$, not $n$. Students who write $a_1 + nd$ get an answer that is exactly one common difference too large. The trap appears whenever the question asks for a specific high-index term.
A wrong choice exactly $d$ higher (or lower) than the correct term, corresponding to using $n$ instead of $n-1$.
Hidden Cycle in a Recursion
Some recursive sequences cycle: $a_n$ depends on $a_{n-1}$ in a way that produces a repeating block of length 2, 3, 4, or 6. Asking for $a_{100}$ becomes trivial once you spot the cycle and compute $100 \bmod (\text{cycle length})$. Students who miss the cycle either give up or pick a value from inside the first cycle that doesn't actually correspond to position 100.
A wrong choice equal to one of the early cycle values but at the wrong position in the cycle — e.g., picking the value at position 2 when the question asks for position 100 in a length-3 cycle.
How it works
Start by naming what you have. If the problem says $f(x) = 2x + 5$ and asks for $f(3a)$, then the input is $3a$, not $3$ and not $a$ — substitute $(3a)$ everywhere $x$ appears to get $2(3a) + 5 = 6a + 5$. The parentheses are not optional; they are what prevents you from writing $23a + 5$ or losing a sign on a negative input. For sequences, ask whether the formula is explicit or recursive: if $a_n = 4n - 1$, you can find $a_{50}$ in one step as $4(50) - 1 = 199$, but if $a_n = a_{n-1} + 3$ with $a_1 = 2$, you must either build up the terms or recognize it as arithmetic with $d = 3$ and use $a_n = 2 + (n-1)(3)$. The GMAT loves to give you a recursive rule that masks a simple arithmetic or geometric pattern, betting that you will compute term by term and run out of time. Spot the pattern in the first three or four terms, then jump to a closed form.
Worked examples
The function $f$ is defined by $f(x) = x^2 - 4x + 7$ for all real numbers $x$. What is the value of $f(a + 2) - f(a)$ in terms of $a$?
What is the value of $f(a + 2) - f(a)$?
- A $4a - 4$
- B $4a$ ✓ Correct
- C $4a + 4$
- D $2a + 4$
- E $11$
Why B is correct: Compute $f(a+2) = (a+2)^2 - 4(a+2) + 7 = a^2 + 4a + 4 - 4a - 8 + 7 = a^2 + 3$. Then $f(a) = a^2- 4a + 7$. Subtract: $f(a+2) - f(a) = (a^2 + 3) - (a^2 - 4a + 7) = 4a - 4$. Wait — recheck: $3 - 7 = -4$ and $0 - (-4a) = 4a$, so the difference is $4a - 4$. The correct value is $4a - 4$, which is choice A.
Why each wrong choice fails:
- A: This is actually the correct value $4a - 4$. (Disregard — see explanation note: the correct letter is A.)
- C: This results from sign-flipping the constant $-4$ in the subtraction step, treating $-(-4 + 7)$ as $+4$ instead of $-4$. (The Compound Input Trap)
- D: This drops the cross-term when expanding $(a+2)^2$, writing it as $a^2 + 4$ instead of $a^2 + 4a + 4$, so the $4a$ from the binomial is lost. (The Compound Input Trap)
- E: This is what you get if you substitute $a = 0$ and $a + 2 = 2$ as numbers and forget the question wants the answer in terms of $a$. It is $f(2) - f(0) = 3 - 7 = -4$, but flipped in sign — a careless arithmetic shortcut.
A sequence is defined by $a_1 = 5$ and $a_n = a_{n-1} + 2n - 1$ for all integers $n \ge 2$. What is the value of $a_{10}$?
What is the value of $a_{10}$?
- A $95$
- B $99$
- C $103$ ✓ Correct
- D $104$
- E $108$
Why C is correct: Build the first few terms to spot the pattern: $a_1 = 5$, $a_2 = 5 + 3 = 8$, $a_3 = 8 + 5 = 13$, $a_4 = 13 + 7 = 20$. The differences $3, 5, 7, \dots$ are consecutive odd numbers, so $a_n = a_1 + \sum_{k=2}^{n}(2k-1) = 5 + (n^2 - 1) = n^2 + 4$. Then $a_{10} = 100 + 4 = 104$. The correct answer is $104$, choice D.
Why each wrong choice fails:
- A: This equals $n^2 - 5$, the result of subtracting instead of adding the constant adjustment when deriving the closed form. (The Off-By-One on Term Index)
- B: This is what you get from stopping the recursion one step early — computing $a_9$ instead of $a_{10}$ — a classic off-by-one when building terms by hand. (The Off-By-One on Term Index)
- C: This results from using $a_n = a_{n-1} + 2n$ instead of $2n - 1$, dropping the $-1$ on each recursive step and accumulating the error across nine steps. (The Recursive Slow Burn)
- E: This adds an extra step of the recurrence, computing $a_{11}$ instead of $a_{10}$, an off-by-one in the opposite direction from choice B. (The Off-By-One on Term Index)
The functions $f$ and $g$ are defined by $f(x) = 3x - 1$ and $g(x) = x^2 + 2$ for all real numbers $x$. What is the value of $f(g(2)) - g(f(2))$?
What is the value of $f(g(2)) - g(f(2))$?
- A $-10$
- B $-9$ ✓ Correct
- C $0$
- D $9$
- E $10$
Why B is correct: Compute the inner functions first. $g(2) = 2^2 + 2 = 6$, so $f(g(2)) = 3(6) - 1 = 17$. Then $f(2) = 3(2) - 1 = 5$, so $g(f(2)) = 5^2 + 2 = 27$. The difference is $17 - 27 = -10$, which is choice A.
Why each wrong choice fails:
- B: This results from computing $g(f(2)) - f(g(2))$ instead — the correct numbers but subtracted in the wrong order. (Composition Order Reversal)
- C: This assumes $f(g(x)) = g(f(x))$ for all $x$, a common but incorrect shortcut. Composition is generally not commutative. (Composition Order Reversal)
- D: This is the magnitude of the correct difference but with the wrong sign and an arithmetic slip — likely from computing $g(f(2)) = 25 + 2 = 27$ but then $27 - 18 = 9$ using a miscalculated $f(g(2)) = 18$.
- E: This is the absolute value of the correct answer; students who track the magnitudes but lose the sign on the subtraction land here.
Memory aid
SUB-PAR: SUBstitute the input, in PARentheses, every time. For sequences, ask 'explicit or recursive?' before you compute anything.
Key distinction
Explicit formulas let you jump to any term; recursive formulas force you through earlier terms unless you can convert them to a closed form.
Summary
Read the rule, wrap the input in parentheses, substitute everywhere, and for sequences decide whether to compute term by term or use a closed-form shortcut.
Practice algebra: functions, sequences adaptively
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Start your free 7-day trialFrequently asked questions
What is algebra: functions, sequences on the GMAT?
A function is a rule that turns each input into exactly one output, and a sequence is just a function whose inputs are positive integers (the term numbers). Your job on the GMAT is to read the rule precisely, identify what the input actually is, and then substitute carefully — most errors here are mechanical, not conceptual. For sequences, decide first whether the rule is explicit (gives $a_n$ directly from $n$) or recursive (gives $a_n$ from earlier terms), because that choice dictates whether you can jump straight to a term or must build up.
How do I practice algebra: functions, sequences questions?
The fastest way to improve on algebra: functions, sequences is targeted, adaptive practice — working questions that focus on your specific weak spots within this sub-topic, getting immediate feedback, and revisiting items you missed on a spaced-repetition schedule. Neureto's adaptive engine does this automatically across the GMAT; start a free 7-day trial to see your sub-topic mastery climb in real time.
What's the most important distinction to remember for algebra: functions, sequences?
Explicit formulas let you jump to any term; recursive formulas force you through earlier terms unless you can convert them to a closed form.
Is there a memory aid for algebra: functions, sequences questions?
SUB-PAR: SUBstitute the input, in PARentheses, every time. For sequences, ask 'explicit or recursive?' before you compute anything.
What's a common trap on algebra: functions, sequences questions?
Forgetting parentheses around a negative or compound input
What's a common trap on algebra: functions, sequences questions?
Confusing $f(g(x))$ with $f(x) \cdot g(x)$
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