PART I · SHORT RESPONSE第一部分 · 短答题SAT-style MCQ + ON/BC short answer · 22 marksSAT 风格选择题 + 安/卑省考短答 · 共 22 分
Section A · Short ResponseA 部分 · 短答题
Mix of multiple-choice and short-answer items. For MCQs, circle the letter; show enough work in the margin that a marker could verify. For short-answer items, state the type of sequence (arithmetic or geometric) and the parameters $(a_1, d)$ or $(a_1, r)$ before substituting. No calculator on Q1–Q4; calculator permitted on Q5.本节包含选择题与短答题。选择题请圈出字母答案，并在空白处写下足以让阅卷人核对的过程。短答题在代入之前，必须先说明数列类型（等差或等比）以及参数 $(a_1, d)$ 或 $(a_1, r)$。Q1–Q4 不可使用计算器；Q5 可用计算器。
A sequence is defined recursively by $a_1 = 4$ and $a_{n+1} = a_n + 6$. Which of the following is the explicit formula for $a_n$?一个数列由递推公式 $a_1 = 4$ 与 $a_{n+1} = a_n + 6$ 定义。下列哪一项是 $a_n$ 的通项公式？
A geometric sequence has first term $a_1 = 5$ and common ratio $r = 2$. Which of the following is $a_8$?一个等比数列的首项为 $a_1 = 5$，公比为 $r = 2$。下列哪一项等于 $a_8$？
(a)State $a_1$ and the common ratio $r$.写出 $a_1$ 与公比 $r$。[1]
(b)Write the explicit formula for the $n$-th term $a_n$.写出第 $n$ 项 $a_n$ 的通项公式。[1]
(c)Use $S_n = a_1 (r^n - 1) / (r - 1)$ to compute the sum of the first $7$ terms. Show each substitution.用 $S_n = a_1 (r^n - 1) / (r - 1)$ 计算前 $7$ 项之和。请逐步代入并展示。[3]
(d)Briefly justify why the formula requires $r \ne 1$.简要说明该公式为何要求 $r \ne 1$。[2]
PART II · EXTENDED RESPONSE第二部分 · 长答题AP-feeder FRQ + honors · 36 marksAP 衔接简答题 + 荣誉级 · 共 36 分
Section B · Extended ResponseB 部分 · 长答题
Show every algebraic step. State the type of sequence (arithmetic or geometric) and the parameters $(a_1, d)$ or $(a_1, r)$ before writing any sum formula. For infinite geometric problems, write $|r| < 1$ before applying $S_\infty = a_1 / (1 - r)$. No calculator on Q6–Q9 unless noted.必须写出每一步代数过程。在使用任何求和公式之前，要先说明数列类型（等差或等比）及参数 $(a_1, d)$ 或 $(a_1, r)$。无穷等比题目须在使用 $S_\infty = a_1 / (1 - r)$ 之前先写出 $|r| < 1$。除非另有说明，Q6–Q9 不可使用计算器。
A theatre has rows of seats arranged so that the front row has $18$ seats, and each subsequent row has $3$ more seats than the row in front of it. The theatre has $20$ rows.某剧院的座位按行排列：第一排有 $18$ 个座位，此后每一排比前一排多 $3$ 个座位。剧院共有 $20$ 排。
(a)Identify $a_1$ and $d$, and write the explicit formula $a_n = a_1 + (n - 1) d$ for the number of seats in row $n$.指出 $a_1$ 和 $d$，并写出第 $n$ 排座位数的通项公式 $a_n = a_1 + (n - 1) d$。[2]
(b)Determine the number of seats in the back row (row $20$).求最后一排（第 $20$ 排）的座位数。[2]
(c)Use the average-times-count form $S_n = \tfrac{n}{2}(a_1 + a_n)$ to compute the total number of seats in the theatre.利用"平均数乘项数"形式 $S_n = \tfrac{n}{2}(a_1 + a_n)$ 求剧院座位总数。[2]
(d)Cross-check your answer using the parameter form $S_n = \tfrac{n}{2}(2 a_1 + (n - 1) d)$. Comment briefly on the equivalence of the two forms (Gauss's pairing argument).再用参数形式 $S_n = \tfrac{n}{2}(2 a_1 + (n - 1) d)$ 核对结果。简要说明两种形式等价的依据（高斯的配对论证）。[2]
(b)Determine the number of terms $n$ in the series.求级数的项数 $n$。[3]
(c)Compute the sum of the series using $S_n = \tfrac{n}{2}(a_1 + a_n)$.用 $S_n = \tfrac{n}{2}(a_1 + a_n)$ 计算级数之和。[2]
(d)State, using function notation, the value of $S(10) - S(5)$ where $S(k)$ denotes the partial sum of the first $k$ terms. Interpret this in one sentence.用函数记号写出 $S(10) - S(5)$ 的值，其中 $S(k)$ 表示前 $k$ 项的部分和。并用一句话解释其含义。[2]
(c)Suppose instead that $S_\infty = 24$. Find the value of $r$.若改设 $S_\infty = 24$，求 $r$ 的值。[3]
(d)Express the repeating decimal $0.\overline{45} = 0.454545\ldots$ as an infinite geometric series, state $a_1$ and $r$, verify $|r| < 1$, and use $S_\infty = a_1/(1-r)$ to write the decimal as a fraction in lowest terms.把循环小数 $0.\overline{45} = 0.454545\ldots$ 表示为无穷等比级数，给出 $a_1$ 与 $r$，验证 $|r| < 1$，再用 $S_\infty = a_1/(1-r)$ 将该循环小数写为最简分数。[4]
Sigma notation packages a series compactly. For each part, identify whether the sum is arithmetic or geometric, then evaluate it in closed form.西格玛符号把级数写成紧凑形式。对每一小题，先判断是等差还是等比求和，然后给出闭式解。
(a)Evaluate $\displaystyle\sum_{k=1}^{30} (2k + 5)$. Identify $a_1$, $a_{30}$, and $d$, then apply the arithmetic sum formula.求 $\displaystyle\sum_{k=1}^{30} (2k + 5)$。先指出 $a_1$、$a_{30}$ 与公差 $d$，再用等差求和公式。[3]
(b)Evaluate $\displaystyle\sum_{k=1}^{6} 4 \cdot 3^{k-1}$. Identify $a_1$ and $r$, then apply the finite geometric sum formula.求 $\displaystyle\sum_{k=1}^{6} 4 \cdot 3^{k-1}$。先指出 $a_1$ 与公比 $r$，再用有限等比求和公式。[3]
(c)Evaluate the infinite sum $\displaystyle\sum_{k=0}^{\infty} 5 \cdot \left(\tfrac{1}{4}\right)^{k}$. First verify that $|r| < 1$, then apply $S_\infty = a_1/(1 - r)$.求无穷和 $\displaystyle\sum_{k=0}^{\infty} 5 \cdot \left(\tfrac{1}{4}\right)^{k}$。先验证 $|r| < 1$，再使用 $S_\infty = a_1/(1 - r)$。[2]
(d)Pull out the constant and split the sum: rewrite $\displaystyle\sum_{k=1}^{10} (7 \cdot 2^{k-1} - 3)$ as $7 \cdot \sum_{k=1}^{10} 2^{k-1} - \sum_{k=1}^{10} 3$, evaluate each piece, and state the total.利用常数提取与求和拆分：把 $\displaystyle\sum_{k=1}^{10} (7 \cdot 2^{k-1} - 3)$ 改写为 $7 \cdot \sum_{k=1}^{10} 2^{k-1} - \sum_{k=1}^{10} 3$，分别求值，再写出总和。[2]
PART III · MODELING / APPLIED第三部分 · 建模 / 应用题Universal · 32 marks通用 · 共 32 分
Section C · Modeling and ApplicationsC 部分 · 建模与应用
Name your variables (with units) before writing equations. Identify the type of sequence (arithmetic or geometric) and the parameters $(a_1, d)$ or $(a_1, r)$ before substituting. Calculator permitted throughout Part III. Round monetary answers to the nearest cent and population answers to the nearest whole.在写方程之前先命名变量并标注单位。代入之前要先指出数列类型（等差或等比）以及参数 $(a_1, d)$ 或 $(a_1, r)$。第三部分全程可使用计算器。货币答案保留到分，人口类答案取整。
Sara deposits CA$2000 into a savings account that earns $4\%$ interest compounded annually. Let $A_n$ denote the balance, in dollars, at the end of year $n$, with $A_0 = 2000$ being the initial deposit.Sara 向一个储蓄账户存入 CA$2000，年利率 $4\%$，按年复利。设 $A_n$ 为第 $n$ 年年末的余额（以美元计），其中 $A_0 = 2000$ 为初始存款。
(a)Argue why $(A_n)$ is a geometric sequence and state $a_1 = A_1$ and the common ratio $r$.论证 $(A_n)$ 是等比数列，并写出 $a_1 = A_1$ 与公比 $r$。[2]
(b)Write the explicit formula $A_n = 2000 \cdot (1.04)^{n}$ and use it to compute the balance after $10$ years.写出通项公式 $A_n = 2000 \cdot (1.04)^{n}$，并用它计算 $10$ 年后的余额。[3]
(c)After how many full years does the balance first exceed CA$3000? Show the inequality you solve.经过几个整年后余额首次超过 CA$3000？请写出所列的不等式。[3]
(d)Briefly state, in one sentence, the connection between this geometric sequence and the exponential function $f(t) = 2000 \cdot (1.04)^{t}$ (Ontario MCR3U C3.1; BC PC12 names the same connection).用一句话简述该等比数列与指数函数 $f(t) = 2000 \cdot (1.04)^{t}$ 之间的联系（Ontario MCR3U C3.1；BC PC12 提到同一联系）。[2]
A medical isotope decays in such a way that each day the amount remaining is $85\%$ of the amount that was present at the start of the previous day. A patient is administered a $40$ mg dose at the start of day $1$. Let $a_n$ denote the amount (in mg) remaining at the start of day $n$, so $a_1 = 40$.某医用同位素的衰变方式为：每一天剩余量是前一天初始量的 $85\%$。某患者在第 $1$ 天初注射 $40$ mg 剂量。设 $a_n$ 为第 $n$ 天初剩余的量（单位 mg），故 $a_1 = 40$。
(a)Argue why $(a_n)$ is a geometric sequence and state the common ratio $r$.论证 $(a_n)$ 是等比数列，并写出公比 $r$。[2]
(b)Write the explicit formula $a_n = a_1 \cdot r^{n-1}$ and use it to compute the amount remaining at the start of day $8$.写出通项公式 $a_n = a_1 \cdot r^{n-1}$，并用其计算第 $8$ 天初剩余的量。[2]
(c)The patient must wait until the amount remaining first drops below $5$ mg before a second dose is given. Solve $a_n < 5$ for the smallest integer $n$.患者必须等到剩余量首次低于 $5$ mg 后才能再次给药。求满足 $a_n < 5$ 的最小整数 $n$。[3]
(d)Describe in one sentence how this geometric sequence is a sample of a continuous exponential-decay function (BC PC12 Curricular Competency: exponential functions to geometric sequences).用一句话说明该等比数列如何是连续指数衰减函数的离散取样（BC PC12 课程能力：指数函数与等比数列的联系）。[2]
(e)State one constraint on $n$ that arises from the situation.写出由情境产生的一个对 $n$ 的限制条件。[1]
An employee accepts a contract that pays CA$30000 in the first year, with each subsequent year's salary increasing by exactly $5\%$ over the previous year. Let $a_n$ denote the salary, in dollars, in year $n$.某员工签订的合同首年薪酬为 CA$30000，此后每一年的薪酬比上一年正好高出 $5\%$。设 $a_n$ 为第 $n$ 年的薪酬（美元）。
(a)Argue why $(a_n)$ is a geometric sequence and state $a_1$ and $r$.论证 $(a_n)$ 是等比数列，并写出 $a_1$ 与公比 $r$。[2]
(b)Write the explicit formula $a_n = a_1 \cdot r^{n - 1}$ and use it to compute the salary in year $10$.写出通项公式 $a_n = a_1 \cdot r^{n - 1}$，并用其计算第 $10$ 年的薪酬。[2]
(c)Use the finite geometric sum $S_n = a_1 (r^n - 1) / (r - 1)$ to compute the total amount earned over the first $10$ years.用有限等比求和公式 $S_n = a_1 (r^n - 1) / (r - 1)$ 计算前 $10$ 年的总收入。[3]
(d)Determine the smallest integer $n$ for which the cumulative total $S_n$ first exceeds CA$500000. Show the inequality you solve.求使累计总和 $S_n$ 首次超过 CA$500000 的最小整数 $n$。请写出所列的不等式。[3]
(e)Briefly compare with the alternative offer of a fixed CA$42000 per year (an arithmetic series with $d = 0$): over the first $10$ years, which offer pays more in total? State the comparison explicitly. 将其与另一份固定年薪 CA$42000（即 $d = 0$ 的等差级数）的合同比较：在前 $10$ 年中，哪一份合同总收入更高？请明确说明比较结果。[2]
🇺🇸 US Common Core美国共同核心HSF-IF.A.3 · HSF-BF.A.2 · HSA-SSE.B.4 · HSF-LE.A.1a · HSF-LE.A.2
🇨🇦 Ontario安大略MPM2D Analytic Geometry / Linear Systems (arithmetic-as-linear groundwork) · MCR3U strand C Discrete Functions (C1.1, C2.2, C3.1) · MHF4U revisitMPM2D 解析几何 / 线性方程组（等差即线性的铺垫）· MCR3U 单元 C 离散函数（C1.1、C2.2、C3.1）· MHF4U 回顾
🇨🇦 British Columbia不列颠哥伦比亚FMP&PC10 Content arithmetic sequences: common difference, first term, general term · PC12 Content geometric sequences and series, infinite geometric series, sigma notationFMP&PC10 内容 等差数列：公差、首项、通项 · PC12 内容 等比数列与级数、无穷等比级数、西格玛符号
Full 3-column Syllabus Map lives in ../Study Guides/Unit_6_Sequences_and_Series.html.完整的三列大纲对照表见 ../Study Guides/Unit_6_Sequences_and_Series.html。