微積分數列問題

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問題一: 假設 \(\{a_n\}\) 為 \(a_ n = n^2 + n\),\(n\)為自然數,且 \(S_n\) 為 \(\{a_n\}\) 的部分總和,例如 \(S_n = \sum_{k=1}^n a_k\)。那麼 \(S_2\) 為多少?
答案為8。

問題二:
假設 \(\{a_n\}\) 為 \(a_1 = 1, a_2 = 1 \) 且 \(a_n = a_{n-1} + a_{n -2}\),對所有 \(n > 2 \) 的自然數,那麼 \(a_6\) 為多少?
答案為8。

問題三:
考慮下面數列:
(1) \(1,3,5,7,9,...\)
(2) 等差數列 \(2,5,8,11,14, ...\)
(3) 等比數列 \(3,9,27,81, ...\)
上述數列中,有幾個數列有正確定義第六項?
答案為2,因為(1)不會知道第六項為多少,但等差與等比數列皆可算出第六項的數。

問題四:
底下何者為等差數列?
A. \( \sqrt{1} , \sqrt{3} , \sqrt{5}, \sqrt{7}, \ldots \)
B. \(11, 22, 33, 44, \ldots \)
C. \( 1, 3, 9, 27, \ldots \)
D. \( -1, 0, 1, -1, 0, 1, \ldots \)
答案為B。

問題五:
若 \(3, a, 13 , b\)為等差級數,那麼a+b為多少?
答案為26。

問題六:
在等差級數\( -110, -106, -102, \ldots \)裡,第幾項會變成正數?
答案為第28項。

問題七:
若一數列 \(\{a_n\}\) 定義為 \(a_1=15, a_{n+1}=a_n+2,\) 且 \(n \in \mathbb{N}.\)。
那麼 \(\{a_n\} \) 的通式為什麼?
答案為\( a_n={2}n+{13} \)

問題八:
此等比數列 \(\{ a_i \}_{i \geq 1} = \{ -7, 21, -63, \ldots, \} \),\(n\)為多少時會得到 \(a_n = 1701\)?
答案為6。

問題九:
若某一等比級數的前 \(n\) 項總和為  \(S_n=3n^2+2n\),那麼第24項值為多少?
答案為143。

問題十:
此數列 \( 1 + 2 + 2^2 + 2^3 + \ldots + 2^{ 14 } \)的值為多少?
答案為 \( 2^{ 15 } - 1\)。

問題十一:
若等比級數的 \(a_4 = 48\) 與 \(a_5 = 96\),那麼前七項的總合為多少?
答案為 762。

問題十二:
此等比級數\( \frac{ 15 } { 2^0} + \frac{ 15 } { 2^1 } + \frac{ 15 } { 2^2 } + \ldots \)和為多少?
答案為 30。

問題十三:
假設\(\displaystyle S_n = \sum_{k=1}^{n} (-1)^{k^2 + k}\),那 \(S_n = \,\)?
答案為\( n \),因為\( k^2 + k \)一定為偶數。

讀書筆記:跟TED學表達,讓世界記住你

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此書所介紹的影片如下:

第一章 釋放內在大師




你腦內的兩個世界


第二章 掌握說故事的技巧






第三章 展開對話





第四章 提供新知










第五章 設計令人談瞠目結舌的橋段



第六章 放輕鬆






第七章 嚴守十八分鐘原則



第八章 運用多重感官體驗









第九章 走自己的路

微積分基礎觀念

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此數列 \( \frac{1}{3^0}, \frac{1}{3^1}, \frac{1}{3^2}, \frac{1}{3^3}, \frac{1}{3^4}, \ldots \) 會趨近於零。

\( \lim_{x \rightarrow 2 } \frac{ x^2 - 4 } { x - 2 } \) 值為 4。

\( \displaystyle \lim_{x \to \infty} \frac {1}{x} \) 值為 0。

此數列 \( \frac{1}{1} + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \frac{1}{9} + \frac{1}{11} + \ldots \) 不會收斂。

\(i^ i \) 為實數(Real Number)。

《創新是一種態度》一書心得與筆記

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本書從如何找出問題的主因談起,到著手解決問題,這一串的過程裡會遇到什麼情況,翟大用了很多例子做解釋。讀完本書,筆者有兩點要記錄下來的心得:

  1. 不論直接或間接,每個人都有自己的思考、習慣等框架在。
  2. 真正的創新通常是集結很多人的力量而出現的。

底下是筆記:
「想解決問題,先突破框架。你想解決的問題,是真正的問題嗎?你一定要吃魚嗎?」


很多人問翟大:台灣企業為什麼都不太能創新?
翟大反問:從小到大,爸媽、老師是不是都叫你要守規矩?


「當你下一次看到一件不合理的事情而去問為什麼時,如果你得到的答案是【向來都是如此】的話,不妨問對方:【請問你是第幾號猴子?】」

「先搞清楚對方是雞還是鴨,再決定你要用什麼話來講,免得雞同鴨講。」

「創新的本體有很多種,每一種都有不同的產生原因和處理方式。一個成功的事業常常是多種創新的結合,千萬不要天真地以為只要有一個小發明,就可以成大功立大業了。」

「很多時候不必強求原創,就像寫出傳頌千古文章的文人,也不會每一句話都是他第一個講出來的。就算你的想法是全世界第一個,難道冰河期之前的文明或是別的星球不會出現過嗎?但你既然要出手,就要有比人強的地方,不然還是不要出來破壞自己的名聲。」

「切記國父孫中山的名言:人生以服務為目的!

「犯了錯引咎辭職,其實不是負責的表現。留下來收拾殘局,再做出一番好成績來補過,才是真負責。」

「把機會成本算上去之後還會賺的項目,才是值得投資的。而員工福利的最終目的,就是減少機會成本。」

「真正的教育改革是很多小事和態度加起來的,而不是一個聳動的大改變。」

「有了點子之後,要從質和量兩個方面來評估可行性。後者可以用數學來進行,前者則要靠豐富的經驗和敏銳的觀察力。你的團隊具備這些能力嗎?」

「如果沒有競爭優勢,再好的點子也只是一個夢。」

玩撲克學代數

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在Youtube上有幾則影片:




算數遊戲

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遊戲一
甲請乙做以下的計算:
  1. 在紙上寫下一個整數。
  2. 將此數加上11後,再乘以2。
  3. 將答案減去20後,再乘以5。
  4. 再將答案減去一開始整數的10倍。
乙做完計算後,甲告訴乙:「答案是10。」

遊戲二
甲請乙做以下的計算:
  1. 隨便想一個整數。
  2. 將此數乘以3。
  3. 將答案加1。
  4. 再將答案乘以3。
  5. 將答案加上最初所想的整數。
若乙最後算出來的答案為96時,那麼最初所想的整數為9。

遊戲三
甲請乙做以下的計算:

  1. 取一個介於11到99之間的數字。
  2. 將此數字加上90。
  3. 去掉百位數後,再加上1。

假設乙回答59,則甲會說乙所選的數字為68 (59+9)。

遊戲四
甲請乙做以下的計算:

  1. 選一個喜歡的正整數。
  2. 加上剛才所選的數字還要大1的數字。
  3. 再加上11。
  4. 再除以2。
  5. 減去原本所想的數字。

最後的答案一定為6。

讀書筆記:大前研一教你立刻交涉成功的商用英語

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英語不是直譯母語
例如需要解釋「為何工作比預計時間還晚完成時」

不好的說法:
Why~~
Explain to me.
Stop it.
Don't do it.
Please turn in this report by Wednesday.

基本上要以「自己」為主語:
I was expecting to receive it yesterday. Can you tell me why that didn't happen?

I would be very happy to wait for another three days, if you have a good reason.

I wouldn't do it.
Not today.
If it's possible, I would like to see this report by Wednesday.


開會時,常用關鍵字
Objective
The objective of this meeting is to decide next year's budget.
The objective of this meeting is to build consensus within the team.
We want to know the main objective of this promotion.

Facilitate
He did well in facilitating the meeting.
She is a great facilitator.
She was very good at facilitating discussions between the two groups.

Clarify
Let me clarify my point. First, ~
Could you clarify what you're proposing?
Could you clarify your point?
We hope to clarify some points before we sign the contract.
I am here to clarify any points that are still unclear.

Reservation
I have reservations about this plan. 我對這項計畫持保留態度
He seems to have reservations about this decision.
We have some serious reservations about the project.

Share
Let me share with you some key factors.
Let me share with you my opinion on this issue.
I would be happy to share my observations on this business plan.

Overview
First, let me give an overview of the project.
Here's an overview of some changes.
Let me go over an overview of this presentation.

Refer
Please also refer to page 30 in the catalogue.
Please refer to the handout for details.
Refer to the following page for further information.

Summarize
Could you summarize what you just said?
The results of the survey can be summarized as follows: ~
In this section, I will summarize the report.

Announce
We are happy to announce the launch of two innovative services.
We expect to announce details later this month.
We regret to announce that our new product has been cancelled.

Launch
Our company launched a new business in Indonesia.
We will launch a new brand in September.
She is now preparing to launch a new career as a photographer.

Release
This model will be released as a limited edition.
We decided to release a new product in November.
The company released some details about their relocation.

Commit
We are commited to improving customer satisfaction.
We are commited to excellence.
We are commited to preserving our environment's precious natural resources.

Aim
We aim to acquire 1 million new users.
The company is aimimg for 20% profitability this year.
We aim to double our revenue by implementing this new strategy.

Generate
We will generate profit.
I'm sure this approach will generate a lot of leads. 我相信這套做法可以開發許多潛在顧客
The company generated 10% more profit compared to last year.
The chart shows a robust growth in profits generated from the company's activities.

Produce
We produce mobile phones.
We will be able to produce more parts by installing the new machine.
This plant produces SUVs.

Advantage
These are the advantages fo proceeding with this plan.
The advantage of thie method is that it is cost-saving.
His plan has the advantage of being less expensive than the other proposals.

Achieve
We achieved a 20% increase in sales year-on-year.
We've achieved a gradual increase in sales.
We are here to help talented workers achieve their full potential.
What do you hope to achieve by the age of 60.

Team
The team worked very hard together and delivered great results.
We have to come together as a team.
Mr. Yamada will join the team next week.

Delegate
I will delegate this project to you.
You have to learn to delegate more.
She did a great job handling the delegated task.
We are making great progress.
The proejct is making steady progress.
We are making progress toward the goal for this fiscal year.

Feedback
Could you give us some feedback on the presentation?
Thank you for your feedback.
I'm glad we got positive feedback from the users.

Performance
The performance of your department is 10% higher than last year.
What can we do to improve our performance?
Could you give us an update on the quarterly performance?
We are seeing strong performance in the mobile application sector.

Result
The annual results show significant improvement in sales.
Our campaign resulted in a big success.
I made 500 calls to companies, which resulted in 25 new leads.

Focus
We will focus on research and development this year.
You need to focus on your task.
We need to focus our efforts on getting the work delivered on time.

Develop
We have to develop a new rating system.
We have to develop a framework that works.
We are constantly developing new ways of simplifying our business process.

Execute
We will execute the plan from Monday.
This restructuring plan will be executed in May.
This strategy will be executed from Monday.
The company executed the strategic plan announced last year.

Promote
Let's promote this service through word of mouth.
Our section is working on a campaign to promote brand awareness.
We were discussing the sales promotion plan for next year.
She got promoted yesterday.

Expand
We intend to expand our business into China.
Our competitor has expanded its market share by 10% over the last year.
We are planning on further expansion.

Challenge
We are facing a big challenge.
That's a challenge.
I like facing challenges.
He challenged an expected notion of business.

Issue
We need to talk about this issue.
We must resolve the issue.
What would you say it the most important issue?

Improve
We can improve our operation.
We have to improve our work efficiency.
We should improve the service based on customer feedback.

Opportunity
I want to take this opportunity to thank everybody.
This is a perfect opportunity to launch the campaign.
This is such a rare opportunity.

Perspective
His advice made me see the issue from a broader perspective.
Let's see this issue from a different perspective.
From my perspective, there is no problem with the analysis.

Consider
Could you consider the other option?
Let's consider other options before we make a decision.
Could you consider my proposals?
Thank you for your consideration.

Monitor
Let's monitor the progress.
Let's monitor the sales before we make a decision.
We monitored the consumer behavior at the drug store.

Revist
We'll revisit this issue next summer.
Let's revisit this topic at a later time.
This issue is worth revisiting.
We're not sure whether this is worth revisiting.

Update
We will keep you updated.
Keep us updated.
Give me an update on the situation.
Please keep me updated on the status of your project.
The data is updated about once every two weeks.

Manage
We managed to complete the task.
He managed to come in time for the meeting.
She manages a department of ten people.
He manages a software development company.

Solve
We will solve this problem.
The problem was easily solved.
What do you think should be done to solve the issue?

Report
I will report on the progress of the project.
He reported on the project to his manager.
I report to Mr. Okawa, our executive vice president.
Who do you report to?

聽不懂對方所說的話時,可用底下說法:
I can't follow you.
I'm confused.

What do you mean?
Could you explain?

透過「PREP」方法讓敘述有層次:
Point: My point is that...
Reason: The reason why I mentioned it is that...
Example: For example...
Point: That's why...

想向對方確認狀況時,可用下列句子:
Let me clarify what you said.
Could you give us more detailed information?
Could elaborate a little bit?
Let me ask some questions.
My I ask some questions?

對聽力沒自信時,可說:
May I ask all of you to speak slowly and clearly?
May I ask you to enunciate?
Please clarify idioms.
Please avoid idioms.

爭取緩衝時間時:
Well, I think have two things to say.
Actually, I had exactly the same idea. However...
Let me think.

不知道對方是哪裡人時,不要用Which country,改用下面的問句:
Where are you from?
Where is your home town?

初次見面場合中,可用帶有正面語意的句子:
I'm very excited to be here.
I was looking forward to coming here.

要指出對方的問題點並給予建議時:
I'm concerned about ...
I'm worried ...
說完後,可再加上:
If you have any requests,...
If you have something you would like me to help you with,...

過去式可用於表達難以開口的事
Do you want some sugar for your coffee? 咖啡要加糖嗎?
Did you want some sugar for your coffee? 您的咖啡是否需要加砂糖?

I wonder if you could work this Saturday.
I wondered if you could work this Saturday.

本來打算這麼做,但是失敗了。
I wanted to ... 的感覺不夠強,可改用I've been wanting...
I've been meaning to ... 一直在想著...

I've been wanting to ... 一直很想要...

指數問題

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\( \LARGE \text{如果} x^{x^{x^{16}}} = 16 \text{,那麼} x^{x^{x^{12}}} \text{會是多少?} \) 將原式改為 \[\LARGE{x^{x^{x^{16}}}=x^{16(\frac{x^{16(\frac{x^{16}}{16})}}{16})}=16}\] 將\(x^{16}\) 以 \(y\) 取代得 \[\LARGE{y^{(\frac{y^{(\frac{y}{16})}}{16})}=16}\] 兩邊同取 \(16\) 次方得 \[\LARGE{y^{y^{(\frac{y}{16})}}=16^{16}}\] 可看出 \(y=16.\) 於是可得\[\LARGE{x^{x^{x^{16}}}=x^{16}=16\Longrightarrow x=16^{\frac{1}{16}}=2^{\frac{1}{4}}}\] \(\LARGE x^{12}=(2^{\frac{1}{4}})^{12}=2^{\frac{12}{4}}=2^3=8\) \(\LARGE x^{x^{12}}=(2^{\frac{1}{4}})^8=2^{\frac{8}{4}}=2^2=4\) \(\LARGE x^{x^{x^{12}}}=(2^{\frac{1}{4}})^4=2^{\frac{4}{4}}=2^1=\boxed{2}\)

讀書心得:蘇菲的世界

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<蘇菲的世界>一書是我在多年前所閱讀的,或許我被此書所影響了也說不定,我總是覺得這世界有種讓我感到很不習慣的地方,例如「為什麼車子通行的道路上要有房子,房子幹嘛不蓋在人行道上。」「為什麼要受教育。」等。

話說哲學類的書籍我已好久沒有看了。

角的分類

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維基百科上可查到個別角的分類有下面幾種:



零角
角度等於0°,或一條線
銳角
角度大於0°且小於90°,或弧度大於0且小於的角。
直角
角度等於90°,或弧度為的角。
鈍角
角度大於90°且小於180°,或弧度大於且小於的角。
平角
角度等於180°,或弧度為的角。
優角或反角
角度大於180°且小於360°,或弧度大於且小於的角。
周角
角度等於360°,或弧度為的角。