5.4 Sequences, Functions Review & Exam Readiness

Arithmetic Sequences

A sequence is an ordered list of numbers (terms). An arithmetic sequence adds the same number — the common difference d — to get from each term to the next. The formula for the nth term is:

aₙ = a₁ + (n − 1)d

where a₁ is the first term.

Worked example. Find the 10th term of 4, 7, 10, 13, … Here a₁ = 4 and d = 3 (each term adds 3). So a₁₀ = 4 + (10 − 1)·3 = 4 + 27 = 31.

Worked example — find d. A sequence has a₁ = 20 and a₅ = 4. Then a₅ = 20 + 4d = 4 ⟹ 4d = −16 ⟹ d = −4, a decreasing sequence: 20, 16, 12, 8, 4.

To find d from any two known terms, subtract consecutive terms (d = a₂ − a₁) — it is constant throughout an arithmetic sequence. A quick test for which type a sequence is: if consecutive differences are equal it is arithmetic; if consecutive ratios are equal it is geometric.

Geometric Sequences

A geometric sequence multiplies by the same number — the common ratio r — each step. The nth term is:

aₙ = a₁·r^(n − 1)

Worked example. Find the 6th term of 3, 6, 12, 24, … Here a₁ = 3 and r = 2. So a₆ = 3·2^(6−1) = 3·32 = 96.

Common mistake: the exponent is (n − 1), not n. The first term uses r⁰ = 1, so plugging n = 1 must return a₁.

Function Behavior Review

ALEKS asks you to read function behavior, not just compute. The high-level facts:

• Polynomials. The leading term controls end behavior. Even degree with positive leading coefficient (like x²) rises on both ends; odd degree (like x³) falls left and rises right. The degree is the highest exponent and caps the number of real zeros.
• Rational functions. A function like f(x) = (x+1)/(x−3) has a vertical asymptote where the denominator is zero (here x = 3) and a horizontal asymptote determined by comparing top and bottom degrees (equal degrees → ratio of leading coefficients).
• Absolute value. |x| is the distance from 0, always ≥ 0, forming a V-shaped graph. Solving |x − 2| = 5 splits into x − 2 = 5 or x − 2 = −5, giving x = 7 or x = −3.

Worked example — domain. The domain of f(x) = 1/(x − 4) is all reals except x = 4, because the denominator cannot be zero. In interval notation: (−∞, 4) ∪ (4, ∞).

Worked example — absolute value. Solve |2x − 1| = 9. Split: 2x − 1 = 9 ⟹ x = 5, or 2x − 1 = −9 ⟹ x = −4. Two solutions: x = 5 and x = −4.

High-Yield Review for Placement

The topics that most often decide your ALEKS placement, in rough order of payoff: solving linear and quadratic equations (factoring and the quadratic formula), simplifying exponents and radicals, working with functions (evaluation, domain, composition), the exponential/log conversions from 5.1, systems of equations, and the trig ratios from 5.3. Geometry of area/volume and word-problem translation round it out. If you can do those reliably, you place into a higher-credit course.

ALEKS Test-Readiness Checklist

ALEKS PPL (by McGraw Hill) is adaptive and open-response — you build answers, there is no multiple choice — and it asks about 25–30 questions, scoring 0–100 as the percent of roughly 314 topics you have mastered. It is placement, not pass/fail, and each college sets its own cutoffs. You get up to 5 attempts with required time in the Prep & Learning Module between them, and only the on-screen ALEKS calculator is allowed (on permitted items). Use these habits to place higher:

1. Work the Prep & Learning Module between attempts — that is where the score gain comes from; ALEKS requires module hours before you may retest.
2. Target topics just above your current level. The module shows what you are "ready to learn" — those adjacent topics raise your score fastest.
3. Don't guess randomly. Open-response means a guess rarely lands; if unsure, skip and let ALEKS adapt rather than mislabeling a topic as mastered.
4. Manage time — about 2–3 hours total. Don't sink 20 minutes into one item; pace across all 25–30 questions.
5. Show no calculator dependence. Practice the special-angle and exponent facts by hand, since the on-screen calculator is limited and unavailable on some items.
6. Retake after real study. Scores are valid roughly 6–12 months; a second attempt after focused module work commonly lifts placement a full course.

Common mistake: treating ALEKS like a one-shot test. The whole design rewards iterative study — students who use the module between attempts routinely place a course higher than their first score.