If you know two terms a_i and a_j (at positions i and j), first find the common difference d = (a_j − a_i) / (j − i). Then write the explicit (nth-term) formula using one of the CED forms: a_n = a_i + d(n − i) (or a_n = a_0 + d n if you know a_0). That gives the value for every whole-number n (sequence’s discrete domain). Quick example: if a_3 = 7 and a_6 = 16, then d = (16−7)/(6−3) = 9/3 = 3. So a_n = a_3 + 3(n−3) = 7 + 3(n−3) = 3n − 2. This aligns with 2.1.A.2–A.3 in the CED (common difference and a_n = a_k + d(n−k)). For more practice and AP-style problems, check the Topic 2.1 study guide (https://library.fiveable.me/ap-pre-calculus/unit-2/change-arithmetic-geometric-sequences/study-guide/TjmiwbtDpN420iuL) and the practice bank (https://library.fiveable.me/practice/ap-pre-calculus).

Arithmetic sequences change by the same amount each step (constant difference d). Use them when a quantity grows or shrinks additively—e.g., monthly savings +$50: a_n = a_0 + dn. Geometric sequences change by the same factor each step (common ratio r). Use them for multiplicative/exponential change—e.g., population or interest that multiplies by 1.05 each period: g_n = g_0·r^n. Key CED points: sequences are functions with a discrete domain (whole numbers), arithmetic has constant rate of change (2.1.A.2–3), geometric has constant proportional change (2.1.B.1–2). Increasing arithmetic adds the same amount each term; increasing geometric adds larger amounts each step because it multiplies (2.1.B.3). On the AP exam, expect discrete-term questions and to write explicit nth-term formulas (showing a0 or g0 and d or r). For a quick review, check the Topic 2.1 study guide (https://library.fiveable.me/ap-pre-calculus/unit-2/change-arithmetic-geometric-sequences/study-guide/TjmiwbtDpN420iuL) and try practice problems (https://library.fiveable.me/practice/ap-pre-calculus).

Think of an arithmetic sequence as a list of numbers that changes by the same amount each step. In the formula a_n = a_0 + d n: - a_n is the nth term of the sequence (the value when you take n steps from the start). - a_0 is the initial value—the term at n = 0 (the “starting” term). AP uses whole-number domain for sequences, so n = 0, 1, 2, ... - d is the common difference: the constant amount you add each time (the sequence’s constant rate of change). - n counts how many steps you moved from the start. Each increase of 1 in n adds exactly d to the term. Example: if a_0 = 5 and d = 3, then a_1 = 5 + 3(1) = 8, a_2 = 5 + 3(2) = 11, etc. You can also use a_k + d(n − k) when your known term isn’t the a_0 term. For more practice and AP-aligned review of Topic 2.1, check the Fiveable study guide (https://library.fiveable.me/ap-pre-calculus/unit-2/change-arithmetic-geometric-sequences/study-guide/TjmiwbtDpN420iuL) and try problems from the unit page (https://library.fiveable.me/ap-pre-calculus/unit-2) or the practice set (https://library.fiveable.me/practice/ap-pre-calculus).

Look at how consecutive terms change. - Arithmetic: successive differences are constant. Subtract each term from the next (a_{n+1} − a_n). If you get the same number every time (the common difference d), it’s arithmetic. Example: 5, 8, 11, 14 → differences 3, 3, 3 so a_n = a_0 + dn. - Geometric: successive ratios are constant. Divide each term by the previous one (g_{n+1} / g_n). If you get the same number every time (the common ratio r), it’s geometric. Example: 2, 6, 18 → ratios 3, 3 so g_n = g_0 r^n. Quick checks and gotchas: - If any term is zero you can still have an arithmetic sequence, but ratio tests fail when dividing by zero. - Ratios can be negative or fractional (alternating signs ok). - Arithmetic changes by equal additive amounts; geometric changes by equal multiplicative factors, so geometric growth usually accelerates. These match the CED definitions (common difference d and common ratio r). For extra practice and quick examples, see the Topic 2.1 study guide (https://library.fiveable.me/ap-pre-calculus/unit-2/change-arithmetic-geometric-sequences/study-guide/TjmiwbtDpN420iuL) and the Unit 2 overview (https://library.fiveable.me/ap-pre-calculus/unit-2). For more problems, try the AP Precalculus practice bank (https://library.fiveable.me/practice/ap-pre-calculus).

For a geometric sequence with common ratio r and an initial term g0, the explicit (nth-term) formula is g_n = g0 * r^n (if you index starting at n = 0). If your sequence is indexed so the kth term is g_k, the formula for the nth term is g_n = g_k * r^(n−k). Key points from the CED: successive terms change by a constant factor (common ratio r), the sequence is a function on whole numbers so the graph is discrete, and this formula is the exponential (multiplicative) model you’ll see on the exam. If your first term is labeled g1 instead of g0, rewrite as g_n = g1 * r^(n−1). For more practice and examples, check the Topic 2.1 study guide (https://library.fiveable.me/ap-pre-calculus/unit-2/change-arithmetic-geometric-sequences/study-guide/TjmiwbtDpN420iuL) and the AP Precalculus Unit 2 overview (https://library.fiveable.me/ap-pre-calculus/unit-2).

The common difference d is just the constant amount you add each step in an arithmetic sequence. Step-by-step: 1. Identify two terms and their positions: say a_k (the kth term) and a_n (the nth term). If you only have consecutive terms a_1, a_2, a_3… you can use any two terms. 2. Use the formula for arithmetic sequences: a_n = a_k + d(n − k). Solve for d: d = (a_n − a_k) / (n − k). 3. If you have consecutive terms a_k and a_{k+1}, then d = a_{k+1} − a_k (quick check). 4. Once you have d and an initial term a_0 (or a_1), write the explicit nth-term: a_n = a_0 + dn (or adjust index: a_n = a_1 + d(n−1)). Example: if a_3 = 11 and a_7 = 27, then d = (27 − 11)/(7 − 3) = 16/4 = 4, so a_n = a_3 + 4(n−3). This matches CED essential knowledge 2.1.A.2–3 (discrete domain, constant rate of change). For more examples and practice, check the Topic 2.1 study guide (https://library.fiveable.me/ap-pre-calculus/unit-2/change-arithmetic-geometric-sequences/study-guide/TjmiwbtDpN420iuL) and try problems on Fiveable’s AP Precalculus practice page (https://library.fiveable.me/practice/ap-pre-calculus).

Use a_n = a_0 + d n when you know the initial term a_0 (the term at n = 0). That formula treats the sequence as a function from whole numbers with index 0 as your starting point (CED 2.1.A.1–2.1.A.3). Use a_n = a_k + d(n − k) when the term you know is not the 0th term but some kth term a_k. This just shifts the same linear rule so the known term is the “anchor.” It’s the right choice whenever your problem gives a term like “the 5th term is 12” (so use a_n = a_5 + d(n−5)) or when the natural indexing in context starts at 1 (you might know a_1 and prefer a_n = a_1 + d(n−1)). Example: common difference d = 3 and a_4 = 10. Then a_n = 10 + 3(n−4). If you converted to a_0 first, a_0 = 10 − 3·4 = −2 and a_n = −2 + 3n (both equivalent). Remember sequences are discrete (whole-number domain). For more practice and AP-aligned examples, check the Topic 2.1 study guide (https://library.fiveable.me/ap-pre-calculus/unit-2/change-arithmetic-geometric-sequences/study-guide/TjmiwbtDpN420iuL) and extra problems (https://library.fiveable.me/practice/ap-pre-calculus).

Think of arithmetic vs geometric as additive vs multiplicative. An arithmetic sequence adds the same amount each step: a_n = a_0 + dn, so every term increases by d. A geometric sequence multiplies by the same factor each step: g_n = g_0 r^n, so each new term is r times the previous one. Example: start at 10. Arithmetic with d = 5: 10, 15, 20, 25, 30 (adds 5 each time). Geometric with r = 1.5: 10, 15, 22.5, 33.75, 50.625 (each step multiplies). Notice the geometric increases get bigger each step because the change itself scales by r. If r>1, the increment grows; if 0https://library.fiveable.me/ap-pre-calculus/unit-2/change-arithmetic-geometric-sequences/study-guide/TjmiwbtDpN420iuL) and try related problems on the Unit 2 page (https://library.fiveable.me/ap-pre-calculus/unit-2).

Think of a salary-increase problem as an arithmetic sequence: each year you add the same amount (the common difference). Follow these steps. 1. Identify the initial term a0 (salary at time 0) and the common difference d (the raise each period). Remember the domain is whole numbers (years, months). 2. Write the explicit formula: a_n = a_0 + d n (or a_n = a_k + d(n−k) if they give a later year). 3. Plug in the whole-number n for the period you want and include units in your answer. Example: start $40,000 with a $2,000 annual raise. a_n = 40000 + 2000n. Salary after 5 years (n=5) is 40000 + 2000(5) = $50,000. On the AP, show your steps (identify a0 and d, write the formula, evaluate) and treat the sequence as discrete points (CED 2.1.A keywords: arithmetic sequence, common difference, a0, explicit formula). For more examples and practice, check the Topic 2.1 study guide (https://library.fiveable.me/ap-pre-calculus/unit-2/change-arithmetic-geometric-sequences/study-guide/TjmiwbtDpN420iuL) and hundreds of practice problems (https://library.fiveable.me/practice/ap-pre-calculus).

Common difference (arithmetic) vs common ratio (geometric): - Common difference (d): an arithmetic sequence adds the same amount each step—a constant additive rate of change. If a0 is the start, a_n = a0 + d·n (or a_n = a_k + d(n−k)). Example: 3, 7, 11, ... has d = 4. On a graph the sequence’s points move up (or down) by equal vertical amounts each term. - Common ratio (r): a geometric sequence multiplies by the same factor each step—a constant proportional (multiplicative) change. If g0 is the start, g_n = g0·r^n (or g_n = g_k·r^(n−k)). Example: 3, 9, 27, ... has r = 3. Geometric sequences grow/shrink by larger amounts each term when r ≠ 1. Why it matters for AP Precalculus: Topic 2.1 distinguishes additive (linear) vs multiplicative (exponential) change; recognize which formula to use and that sequence graphs are discrete. For more examples and practice, check the Topic 2.1 study guide (https://library.fiveable.me/ap-pre-calculus/unit-2/change-arithmetic-geometric-sequences/study-guide/TjmiwbtDpN420iuL) and tons of practice problems (https://library.fiveable.me/practice/ap-pre-calculus).

Take any two consecutive terms and divide the later term by the earlier term—that quotient is the common ratio r. In symbols, for a geometric sequence g_n, r = g_n / g_{n-1} (or r = g_k+1 / g_k for any k). Always use consecutive terms and simplify the fraction exactly (don’t round on the AP). Example: terms 3/4, 1/2, 1/3. Compute r = (1/2) ÷ (3/4) = (1/2)·(4/3) = 2/3. Check: (1/3) ÷ (1/2) = (1/3)·2 = 2/3, so r = 2/3 for the whole sequence. If terms are negative, r can be negative (e.g., 1/2, −1/4 gives r = −1/2). If a term is 0, you can’t divide by it; that breaks a standard geometric sequence. On the AP exam give the exact fractional r when possible (it’s part of expressing geometric sequences as g_n = g_0·r^n). For more examples and practice, see the Topic 2.1 study guide (https://library.fiveable.me/ap-pre-calculus/unit-2/change-arithmetic-geometric-sequences/study-guide/TjmiwbtDpN420iuL) and the AP Precalculus practice pool (https://library.fiveable.me/practice/ap-pre-calculus).

Good question—because sequences are functions whose inputs are whole numbers, their domain is discrete, not all real numbers. The CED says a sequence is a function from the whole numbers to the reals (2.1.A.1), so a_n (or g_n) is only defined at n = 0,1,2,... (for example a_n = a_0 + dn or g_n = g_0 r^n). That means you only get ordered pairs like (0,a_0), (1,a_1), (2,a_2), etc. There’s no value specified for x = 1.5 or x = 2.7, so you can’t legitimately draw a continuous curve between the dots—connecting them would imply the sequence gives values for non-integer inputs, which it doesn’t. (If you do want a continuous graph, you’d have to define an interpolation or an analytic function on all reals.) For more practice and the CED-aligned explanation, check the Topic 2.1 study guide (https://library.fiveable.me/ap-pre-calculus/unit-2/change-arithmetic-geometric-sequences/study-guide/TjmiwbtDpN420iuL) and try practice problems (https://library.fiveable.me/practice/ap-pre-calculus).

Short memory tricks: arithmetic = add, geometric = multiply. - Arithmetic: successive terms change by a constant difference d (constant rate of change). Explicit formula: a_n = a_0 + d n (or a_n = a_k + d(n − k)). Example: 2, 5, 8, 11 → d = 3 so a_n = 2 + 3n. - Geometric: successive terms change by a constant ratio r (constant proportional change). Explicit formula: g_n = g_0 r^n (or g_n = g_k r^(n−k)). Example: 3, 6, 12, 24 → r = 2 so g_n = 3·2^n. Quick checks on problems: subtract consecutive terms—if differences are constant, it’s arithmetic; divide consecutive terms—if ratios are constant, it’s geometric. Remember AP wording: “common difference” = arithmetic, “common ratio” or “multiplicative change” = geometric. Graphs use a discrete domain (whole numbers)—arithmetic looks linear in the points, geometric looks exponential (changes grow/shrink multiplicatively). For practice and AP alignment (Topic 2.1), see the study guide (https://library.fiveable.me/ap-pre-calculus/unit-2/change-arithmetic-geometric-sequences/study-guide/TjmiwbtDpN420iuL) and extra problems (https://library.fiveable.me/practice/ap-pre-calculus).

Look for whether each step adds/subtracts the same amount (arithmetic) or multiplies/divides by the same factor (geometric). Quick checklist: - If the context says “adds 5 people every week,” “decreases by 2 each day,” or “gain/lose a fixed number” → arithmetic. Use a_n = a_0 + d·n (common difference d). - If it says “doubles each period,” “grows by 10% per month,” or “multiplied by 3/4 each step” → geometric. Use g_n = g_0·r^n (common ratio r). - With data, compute successive differences and successive ratios: constant differences → arithmetic; constant ratios → geometric. - Use units: arithmetic change gives a constant rate (units/day etc.), geometric change gives constant proportional change (percent per period). Remember sequences are discrete (domain = whole numbers) on the AP CED (2.1.A and 2.1.B). If you want examples and practice problems, check the Topic 2.1 study guide (https://library.fiveable.me/ap-pre-calculus/unit-2/change-arithmetic-geometric-sequences/study-guide/TjmiwbtDpN420iuL) and the Unit 2 overview (https://library.fiveable.me/ap-pre-calculus/unit-2). For extra practice, see Fiveable’s AP Precalculus practice set (https://library.fiveable.me/practice/ap-pre-calculus).

Saying “a sequence is a function from the whole numbers to the real numbers” means each term of the sequence is just the function’s output when you plug in a whole number (n = 0,1,2,3,...). So a sequence a_n is like a rule f(n) whose domain is the discrete set of whole numbers, and its outputs are real numbers. Important consequences for AP Precalculus (CED 2.1.A.1): the graph is a bunch of isolated points, not a continuous curve, and you should treat n as an integer step counter when using formulas like a_n = a_0 + d n (arithmetic) or g_n = g_0 r^n (geometric). On the exam, expect sequence questions to use discrete-domain thinking (identify common difference or ratio, write explicit nth-term). For extra practice and quick review, see the Topic 2.1 study guide (https://library.fiveable.me/ap-pre-calculus/unit-2/change-arithmetic-geometric-sequences/study-guide/TjmiwbtDpN420iuL) and more practice problems (https://library.fiveable.me/practice/ap-pre-calculus).