IVE Foundation Mathematics I (ITE3705 / GCNU304A)

Tutorial Exercise Set B: Arithmetic and Geometric Sequences

課堂練習工作紙 B：等差與等比數列

---

Questions (題目部分)

Question 1

A student tracks their monthly MTR concession savings in Hong Kong dollars: $14,23,32,41,\dots$14,23,32,41,….

一名同學記錄其每月獲得的港鐵特惠乘車優惠（港幣）：$14,23,32,41,\dots$14,23,32,41,…。

Write down the common difference $d$d of this arithmetic progression.

寫出該等差數列的公差 $d$d。

---

Question 2

In an anime fantasy adventure, an adventurer’s stamina points deplete per battle stage according to an arithmetic progression: $95,87,79,71,\dots ,7$95,87,79,71,…,7.

在某奇幻冒險動漫中，冒險者的體力值隨戰鬥關卡依等差數列遞減：$95,87,79,71,\dots ,7$95,87,79,71,…,7。

Find the number of terms in this arithmetic progression.

求該等差數列的項數。

---

Question 3

Find the arithmetic mean between $-24$−24 and $-8$−8.

求 $-24$−24 與 $-8$−8 之間的等差中項。

---

Question 4

A display technician at the Hong Kong Convention and Exhibition Centre needs to insert three arithmetic means between 7 and 39 to place support rods evenly.

一名在香港會議展覽中心工作的佈置技術員，需要在 7 與 39 之間插入三個等差中項以均勻安裝支撐架。

Insert three arithmetic means between 7 and 39.

在 7 與 39 之間插入三個等差中項。

---

Question 5

Find the sum of the following arithmetic series:

計算下列等差級數的總和：

(a) $2+7+12+\dots$2+7+12+… to 25 terms.

(a) $2+7+12+\dots$2+7+12+… 至第 25 項之和。

(b) $90+84+78+\cdots +30$90+84+78+⋯+30.

(b) $90+84+78+\cdots +30$90+84+78+⋯+30。

---

Question 6

For the arithmetic sequence $y-10,y-4,y+2,y+8,\dots$y−10,y−4,y+2,y+8,…, find:

已知等差數列 $y-10,y-4,y+2,y+8,\dots$y−10,y−4,y+2,y+8,…，求：

(a) the common difference $d$d.

(a) 公差 $d$d。

(b) the general term $T(n)$T(n) expressed in terms of $y$y and $n$n.

(b) 以 $y$y 及 $n$n 表示的通項 $T(n)$T(n)。

(c) the 15th term $T(15)$T(15).

(c) 第 15 項 $T(15)$T(15)。

---

Question 7

For the arithmetic series $5+11+17+\dots$5+11+17+…, find:

已知等差級數 $5+11+17+\dots$5+11+17+…，求：

(a) the 12th term $T(12)$T(12).

(a) 第 12 項 $T(12)$T(12)。

(b) the 25th term $T(25)$T(25).

(b) 第 25 項 $T(25)$T(25)。

(c) the sum of the series from the 12th term to the 25th term inclusive.

(c) 該級數由第 12 項至第 25 項（包含兩端）的總和。

---

Question 8

A local logistics center in Kwai Chung stacks cargo crates in a trapezoidal pattern where the bottom layer has 40 crates, the second layer from the bottom has 37 crates, the third layer has 34 crates, and so on.

葵涌某物流中心將貨物箱堆疊成梯形，底層有 40 箱，倒數第二層有 37 箱，倒數第三層有 34 箱，依此規律遞減。

There are 12 layers of crates in total, and the warehouse supervisor removes the uppermost 5 layers to load onto delivery trucks.

這堆貨物共有 12 層，倉庫主管將最頂層的 5 層貨物搬上貨車運走。

Find:

求：

(a) the total number of crates in the whole stack.

(a) 整堆貨物箱的總數量。

(b) the number of crates loaded onto the delivery trucks (the uppermost 5 layers).

(b) 搬上貨車的貨物箱總數量（即最頂部的 5 層）。

---

Question 9

In an online battle arena game, a player’s attack speed multiplier follows a geometric progression.

在某款多人線上競技遊戲中，角色的攻擊速度加成數值符合等比數列。

Find the first term $a$a of the geometric progression if the 6th term is 96 and the common ratio is 2.

若該等比數列的第 6 項為 96，公比為 2，求其首項 $a$a。

---

Question 10

A technician’s annual salary increases by 8% each year after joining an engineering firm in Kowloon Bay.

一名技術員入職九龍灣一間工程公司後，年薪每年按固定比例遞增 8%。

His annual salary in the year 2018 was HK$180,000.

他在 2018 年的年薪為 180,000 港元。

Find his total earnings from 2018 to 2024 inclusive (correct to the nearest dollar).

求他由 2018 年至 2024 年（包含首尾兩年）的總收入（答案準確至最接近的元）。

---

Question 11

Given the geometric series $5+15+45+\dots$5+15+45+….

已知等比級數 $5+15+45+\dots$5+15+45+…。

If the sum of the first $n$n terms of the series is 5465, find the value of $n$n.

若該級數首 $n$n 項之和為 5465，求 $n$n 的數值。

---

Question 12

For the geometric series $2-6+18-54+\cdots -1458$2−6+18−54+⋯−1458, find:

已知等比級數 $2-6+18-54+\cdots -1458$2−6+18−54+⋯−1458，求：

(a) the number of terms in the series.

(a) 該級數的項數。

(b) the sum of the series.

(b) 該級數的總和。

---

Question 13

The 2nd and 5th terms of a geometric progression are $\frac{8}{3}$38​ and 72 respectively.

某等比數列的第 2 項及第 5 項分別為 $\frac{8}{3}$38​ 與 72。

Find:

求：

(a) the common ratio $R$R.

(a) 公比 $R$R。

(b) the first term $a$a.

(b) 首項 $a$a。

---

Question 14

In a fantasy battle simulation, an aura heals $16\text{ HP}$16 HP on tick 1, $4\text{ HP}$4 HP on tick 2, $1\text{ HP}$1 HP on tick 3, and continues indefinitely with a constant ratio.

在某奇幻戰鬥模擬中，光環在第 1 次跳動時恢復 16 點生命值，第 2 次恢復 4 點，第 3 次恢復 1 點，並以固定公比無限持續。

Find the sum to infinity $S(\mathrm{\infty })$S(∞) of the geometric series $16+4+1+\dots$16+4+1+….

求等比級數 $16+4+1+\dots$16+4+1+… 的無限項之和 $S(\mathrm{\infty })$S(∞)。

---

Question 15

An investor compares two short-term wealth management plans offered by Hong Kong banks for 1 year:

一名投資者正比較香港兩間銀行推出為期 1 年的短期理財方案：

• Plan Alpha (阿爾法方案): Deposit HK$2,000 on the first day of each month for 12 months. Interest is compounded monthly at a rate of 0.5% per month for 1 year.

每月首天存入 2,000 港元，連續存入 12 個月，按月複利計息，月利率為 0.5%，期滿 1 年。

• Plan Beta (貝塔方案): Deposit HK$12,000 on 1 January and HK$12,000 on 1 July. Interest is compounded half-yearly at a rate of 8% p.a. for 1 year.

於 1 月 1 日存入 12,000 港元，並於 7 月 1 日再存入 12,000 港元，按半年複利計息，年利率為 8%，期滿 1 年。

Compare the accrued amounts at the end of 1 year and determine which plan offers a better return.

比較 1 年後的本利和總額，並指出哪一個方案能帶來較高回報。

---

Question 16

An IVE graduate borrows HK$600,000 from a financial institution to set up an AI design studio.

一名 IVE 畢業生向金融機構貸款 600,000 港元以創立一間人工智能設計工作室。

He repays HK$$x$ to the institution on the last day of each month.

他在每月的最後一天向機構等額償還 $x$x 港元。

At an interest rate of 4.8% per annum compounded monthly, the loan will be completely cleared after 5 years.

在年利率為 4.8% 且按月複利計息的條件下，該筆貸款將於 5 年後完全還清。

Find the monthly payment value of $x$x (correct to 2 decimal places).

求每月還款額 $x$x 的數值（答案準確至小數點後兩位）。

---