2021-12-01

My English Words List - December - 2021

endemic

adjective

COVID-19 is likely to become an endemic disease.

A Nature survey shows many scientists expect the virus that causes COVID-19 to become endemic, but it could pose less danger over time.

Eighteen months later, most leading immunologists believe Covid-19 will become endemic—a persistent but manageable threat on par with seasonal flu—conceivably by the end of 2022.

Covid will become an endemic disease as early as 2024, Pfizer executives said Friday, meaning the virus will transition from a global emergency to a constant presence causing regional outbreaks across the world — much like the flu.

noun

It typically takes a few years for a new viral pathogen to move from pandemic to endemic

omicron

noun

the 15th letter of the Greek alphabet

In November 2021, a COVID-19 variant was named Omicron, after the 15th letter of the Greek alphabet.

Omicron

instant

adjective

instant coffee

Is this coffee instant or regular?

Instant coffee

Instant noodle

Instant Pot

clementine

noun

jicama

noun

Pachyrhizus erosus

paprika

noun

Mallorcan pimentón tap de cortí paprika

tomato sauce made with garlic, paprika, and pepper

Paprika

Worcestershire sauce

noun

Worcestershire sauce

kale

noun

Kale

bunch

noun

a bunch of grapes

a bunch of friends

a bunch of children

a bunch of money

He always had a bunch of keys on his belt.

Dried herbs hung in bunches from the kitchen rafters.

parchment

noun

Ancient people wrote on parchment.

Archimedes Palimpsest

recipe

noun

This is one of my grandmother’s recipes.

Recipe

palindrome

noun

The word “dad” and the number “1881” are palindromes.

Palindrome

cartridge

noun

a case or container that holds a substance, device, or material which is difficult, troublesome, or awkward to handle and that usually can be easily changed

The printer needs a new ink cartridge.

Ink cartridge

Cartridge (firearms)

clipper

noun

a device used for cutting something

hair clippers

nail clippers

Hair clipper

Nail clipper

screwdriver

noun

Screwdriver

screw

noun

An assortment of screws

Screw

verb

Remember to screw the lid back on the jar.

Screw the light bulb into the fixture.

Screw the cap on tight.

The lid screws onto the jar.

waiver

noun

Please read the following waivers and agreements carefully.

a criminal defendant’s waiver of a jury trial

The college got a special waiver from the town to exceed the building height limit.

He signed an insurance waiver before surgery.

Waiver

waive

verb

waive a jury trial

waive the fee

She waived her right to a lawyer.

The university waives the application fee for low-income students.

The bank manager waived the charge (= said we didn’t have to pay), as we were old and valued customers.

toad

noun

toad

broom

noun

durian

noun

Durian

aisle

noun

The bride walked down the aisle to the altar.

Aisle

yell

verb

to make a sudden, loud cry

We saw people yelling for help.

I heard someone yelling my name.

billiards

noun

Cue sports

stapler

noun

Stapler

awkward

adjective

an awkward attempt

an awkward design

She is awkward at dancing.

The machine is very awkward to operate.

passage

noun

Her office is at the end of the passage.

the passage of food through the digestive system

the passage of air into and out of the lungs

the passage from life to death

Hallway

kiddo

noun

child, kid
used by an adult to speak to a young person

reverence

noun

Their religion has/shows a deep reverence for nature.

Reverence for or worship of the dead is found in all societies, because belief in life after death is universal. — World Religions, 1983

Reverence for Life says that the only thing we are really sure of is that we live and want to go on living. This is something that we share with everything else that lives, from elephants to blades of grass—and, of course, every human being. So we are brothers and sisters to all living things, and owe to all of them the same care and respect, that we wish for ourselves.

Reverence for Life

vestibule

noun

please leave your wet boots in the vestibule

Synonyms

entranceway, entry, entryway, hall, hallway, lobby

Vestibule (architecture)

incisor

noun

a tooth (as any of the four front teeth of the human upper or lower jaw) for cutting

Permanent teeth of the right half of the lower dental arch, seen from above.

canine

noun

a conical pointed tooth especially : one situated between the lateral incisor and the first premolar

molar

noun

a large tooth near the back of the mouth with a broad surface used for grinding

ditch

verb

ditch an old car

If you are using a single-layer cloth mask to protect yourself from a new COVID-19 variant that is believed to be more than six times as infectious as Delta “ditch it, full stop.”

Anglosphere

noun

the countries of the world in which the English language and cultural values predominate

Anglosphere

platter

noun

large platters of hot turkey and ham

She ordered the seafood platter.

Platter (dishware)

oat

noun

Oat

earbud

noun

a small earphone inserted into the ear

Headphones

bud

noun

The bush has plenty of buds but no flowers yet.

Bud

verb

The trees budded early this spring.

canonical

adjective

As a canonical example

His proposals were generally accepted as canonical.

gospel

noun

a reading from the Gospel of St. John

Gospel

drizzle

Drizzle

noun

rain that falls lightly in very small drops

Yes, it’s raining, but it’s only a drizzle.

the intermittent drizzle was just heavy enough to spoil all of our outdoor activities

verb

to rain in very small drops

It was beginning to drizzle, so she pulled on her hood.

rebate

noun

There is a $50 rebate offered with the printer.

Mail in your receipt and get a rebate.

Rebate (marketing)

Tax refund

wrestle

verb

They’ll be wrestling each other for the championship.

good-for-nothing

adjective

of no use or value

sparrow

noun

a common type of small bird that usually has brown or gray feathers

Old World sparrow

New World sparrow

parrot

noun

Parrot

verb

Some of the students were just parroting what the teacher said.

the toddler parroted everything her father said, often to the latter’s embarrassment

biodegradable

adjective

biodegradable trash bags

fasten

verb

Make sure the lid is tightly fastened.

two boards fastened together by nails

fastener

Hook-and-loop fastener

trampoline

noun

Trampoline

2021-11-29

Primes

by Grant Sanderson

The primes,
through times,

mystified
those who pried.

One fact answers why
they’re simple yet sly:

Layers of abstraction yield
complex forms when pierced and peeled.

Addition lies under multiplication,
defining him as repeated summation.

Then he defines primes as the atoms of integers,
for when multiplied, they give numbers their signatures.

But when we breach the layer between these two distinct operations,
asking about how primes add and subtract, there are endless frustrations.

Even innocuous questions, “what are all their sums?”, or “how often do they differ by two?”,
stump everyone who has ever lived, with progress made only quite recently by just a few.

However, to recruit for and progress math we need to have such questions which can be phrased simply and remain unsolved.
What child does not hear such conjectures and dream, if only for a moment, that they will be the one to see them resolved?

For otherwise the once vigorous curiosity of a child towards math’s patterns, as they grow older, tends to grow tame,
just as the rhyme and rhythm of the primes seems to fade as numbers grow, though in both the underlying patterns remain the same.

Math Poetry
20 is not a prime number, 21 is not a prime number, 2021(=43x47) is not a prime number.
11 is a prime number, 29 is a prime number, 1129 is a prime number.

2021-11-25

Congruent Number and Elliptic Curve

Once Marcus du Sautoy asked Sir Andrew Wiles what he’d scribble in the margin he’d proved to tease next generation. Answer: congruent number problem

A congruent number is a positive integer that is the area of a right triangle with three rational number sides.

$
\displaystyle
\begin{matrix}
a^{2}+b^{2}&=&c^{2},\\
\frac{1}{2}ab&=&n.
\end{matrix}
$

$
\displaystyle
\begin{matrix}
a^2 = c^2 - b^2 \\
2a^2 + 2ac = a^2 + c^2 + 2ac - b^2 \\
2ab(a+c) = b(a+c)^{2} - b^{3} \\
4n(a+c) = b(a+c)^{2}-b^{3} \\
\frac{4n(a+c)}{b^4} = \frac{(a+c)^{2}}{b^{3}}-\frac{1}{b} \\
\frac{4n^{4}(a+c)^{2}}{b^{4}} = \frac{n^{3}(a+c)^{3}}{b^3}-\frac{n^3(a+c)}{b} \\
(\frac{2n^2(a+c)^2}{b^2})^{2} = (\frac{n(a+c)}{b})^{3} - n^{2}(\frac{n(a+c)}{b}) \\
set \hspace{1mm} y = \frac{2n^2(a+c)}{b^2}, \\
x = \frac{n(a+c)}{b} \\
y^2 = x^3 - n^2x
\end{matrix}
$

n is a congruent number if and only if the elliptic curve $ y^2 = x^3-n^2x $ contains a rational point with $ y \neq 0 $

$
\displaystyle
\begin{matrix}
x = \frac{n(a+c)}{b} = \frac{n(c+a)(c-a)}{b(c-a)} = \frac{nb}{c-a} \\
y = \frac{2n^2(a+c)}{b^2} = \frac{2n^2(a+c)}{c^2-a^2} = \frac{2n^2}{c-a} \\
\frac{x}{y}=\frac{b}{2n} \\
b = \frac{2nx}{y}, \\
a=\frac{2a^2+2ac}{2(a+c)} \\
=\frac{a^2+2ac+a^2}{2(a+c)} \\
=\frac{a^2+2ac+c^2-b^2}{2(a+c)} \\
=\frac{(a+c)^2-b^2}{2(a+c)} \\
=\frac{n^2(a+c)^2-b^2n^2}{2n^2(a+c)} \\
=\frac{(\frac{n(a+c)}{b}^{2}-n^2}{\frac{2n^2(a+c)}{b^2}} \\
=\frac{x^2-n^2}{y}, \\
c=\frac{2c(a+c)}{2(a+c)} \\
=\frac{2c^2+2ac}{2(a+c)} \\
=\frac{a^2+c^2+2ac+c^2-a^2}{2(a+c)} \\
=\frac{(a+c)^2+c^2-a^2}{2(a+c)} \\
=\frac{(a+c)^2+b^2}{2(a+c)} \\
=\frac{n^2(a+c)^2+b^2n^2}{2n^2(a+c)} \\
=\frac{(\frac{n(a+c)}{b})^2+n^2}{\frac{2n^2(a+c)}{b^2}} \\
=\frac{x^2+n^2}{y}
\end{matrix}
$

We have a bijection(one-to-one correspondence)

$
\Large
\{(a,b,c) \in \mathbb{Q}^{3} \mid a^2+b^2=c^2,\frac{ab}{2}=n \} \leftrightarrow \{(x,y) \in \mathbb{Q}^{2} \mid y^2 = x^3-n^2x \hspace{1mm} and \hspace{1mm} y \neq 0 \}
$

with inverse functions

$
\Large
(a,b,c) \mapsto (\frac{nb}{c-a},\frac{2n^2}{c-a}) \hspace{8pt} and \hspace{8pt} (x,y) \mapsto (\frac{x^2-n^2}{y},\frac{2nx}{y},\frac{x^2+n^2}{y})
$

$ y^2 = x^3-25x $ has rational point $ x=-4, y= \pm 6 $, 5 is a congruent number.

$
\begin{align*}
\begin{cases}
a = \frac{-4^2-5^2}{-6} = \frac{3}{2} \\
b = \frac{2\times5\times-4}{-6} = \frac{20}{3} \\
c = \frac{-4^2+5^2}{6} = \frac{41}{6}
\end{cases}
\end{align*}
$

$ y^2 = x^3-36x $ has rational points
$
\displaystyle
\begin{matrix}
x=-3,\hspace{1mm} y= \pm 9, \\
x=-2,\hspace{1mm} y= \pm 8, \\
x=12,\hspace{1mm} y= \pm 36, \\
x=18,\hspace{1mm} y= \pm 72, \\
\end{matrix}
$,

6 is a congruent number.

$
\begin{align*}
\begin{cases}
a = \frac{12^2-6^2}{36} = 3 \\
b = \frac{2\times6\times12}{36} = 4 \\
c = \frac{12^2+6^2}{36} = 5
\end{cases}
\end{align*}
$

One very old problem concerned with rational points on elliptic curves is the congruent number problem. One way of stating it is to ask which rational integers can occur as the areas of right-angled triangles with rational length sides. Such integers are called congruent numbers. … It is closely related to the problem of determining the rational points on the curve $ C_n : y^2 = x^3 − n^2x $. … As an example of this, consider the conjecture of Euler from 1769 that $ x^4 + y^4 + z^4 = t^4 $ has no non-trivial solutions. By finding a curve of genus 1 on the surface and a point of infinite order on this curve, Noam Elkies found the solution $ 2682440^4 + 15365639^4 + 18796760^4 = 20615673^4$ . His argument shows that there are infinitely many solutions to Euler’s equation. – The Birch And Swinnerton-Dyer Conjecture, by Andrew Wiles
Assuming the BSD Conjecture and using Tunnell’s theorem, we conclude that $ 1234567 = 127 \times 9721, 1234567 \equiv 7 \hspace{2mm} (mod \hspace{1mm} 8) $ is indeed a congruent number. (Chapter 15 The Conjecture of Birch And Swinnerton-Dyer, 6 The Congruent Number Problem, Elliptic Tales: Curves, Counting, and Number Theory, by Avner Ash and Robert Gross)

2021-11-19

Managing Oneself - Harvard Business Review - January 2005 issue

by Peter F. Drucker

Success in the knowledge economy comes to those who know themselves—their strengths, their values, and how they best perform.

Summary. Throughout history, people had little need to manage their careers—they were born into their stations in life or, in the recent past, they relied on their companies to chart their career paths. But times have drastically changed. Today we must all learn to manage ourselves.

What does that mean? As Peter Drucker tells us in this seminal article first published in 1999, it means we have to learn to develop ourselves. We have to place ourselves where we can make the greatest contribution to our organizations and communities. And we have to stay mentally alert and engaged during a 50-year working life, which means knowing how and when to change the work we do.

It may seem obvious that people achieve results by doing what they are good at and by working in ways that fit their abilities. But, Drucker says, very few people actually know—let alone take advantage of—their fundamental strengths.

He challenges each of us to ask ourselves: What are my strengths? How do I perform? What are my values? Where do I belong? What should my contribution be? Don’t try to change yourself, Drucker cautions. Instead, concentrate on improving the skills you have and accepting assignments that are tailored to your individual way of working. If you do that, you can transform yourself from an ordinary worker into an outstanding performer.

Today’s successful careers are not planned out in advance. They develop when people are prepared for opportunities because they have asked themselves those questions and have rigorously assessed their unique characteristics. This article challenges readers to take responsibility for managing their futures, both in and out of the office.

History’s great achievers—a Napoléon, a da Vinci, a Mozart—have always managed themselves. That, in large measure, is what makes them great achievers. But they are rare exceptions, so unusual both in their talents and their accomplishments as to be considered outside the boundaries of ordinary human existence. Now, most of us, even those of us with modest endowments, will have to learn to manage ourselves. We will have to learn to develop ourselves. We will have to place ourselves where we can make the greatest contribution. And we will have to stay mentally alert and engaged during a 50-year working life, which means knowing how and when to change the work we do.

What Are My Strength?

Most people think they know what they are good at? They are ususlly wrong.

The only way to discover your strength is through feedback analysis.

How Do I Perform?

Am I a reader or a listener?

How do I learn?

What Are My Values?

I call it the “mirror test”.

Where Do I Belong?

What Should I Contribute?

To answer it, they must address three distinct elements:

What does the situation require?
Given my strength, my way of performing, and my values, how can I make the greatest contribution to what needs to be done?
And finally, What results have to be achieved to make a difference?

Responsibility for Relationships

The first is to accept the fact that other people are as much individuals as you youself are.
The second part of relationship responsibility is taking responsibility for communication.

The Second Half of Your Life

There are three ways to develop a second career:

The first is actually to start one. We will see many more second careers undertaken by people who have achived modest success in their first jobs.
The second way to prepare for the second half of yours life is to develop a parallel career.
Fially, there are the social entrepreneurs.

In effect, managing oneself demands that each knowledge worker think and behave like a chief executive office.

Knowledge workers outlive their organizations, and they are mobile. The need to manage oneself is therefore creating a revolution in human affairs.

Peter F. Drucker is the Marie Rankie Clarke Professor of Social Science and Management(Emeritus) at Claremont Graduate University in Claremont, California. This article is an excerpt from his book Management Challenges for the 21st Century(HarperCollins, 1999)

Managing Oneself

2021-11-10

Scientific method & Debugging

Scientific method

Observe some feature of the natural world, generally with precise measurements.
Hypothesize a model that is consistent with the observations.
Predict events using the hypothesis.
Verify the predictions by making further observations.
Validate by repeating until the hypothesis and observations agree.

One of the key tenets of the scientific method is that the experiments we design must be reproducible, so that others can convince themselves of the validity of the hypothesis. Hypotheses must also be falsifiable, so that we can know for sure when a given hypothesis is wrong (and thus needs revision). As Einstein famously is reported to have said (“No amount of experimentation can ever prove me right; a single experiment can prove me wrong”), we can never know for sure that any hypothesis is absolutely correct; we can only validate that it is consistent with our observations.

Scientific Debugging

Observe. we look at the behavior of the program. What are its outputs? What information is it displaying? How is it responding to user input?
Hypothesize. we try to divide the set of possible causes into multiple independent groups. For example, in a client/server app, a potential hypothesis might be “the bug is in the client”, which is either true or false; if false, then presumably the bug is on the server. A good hypothesis should be falsifiable, which means that you ought to be able to invent some test which can disprove the proposition.
Experiment. we run the test — that is, we execute the program in a way that allows us to either verify or invalidate the hypothesis. In many cases, this will involve writing additional code that is not a normal part of the program; you can think of this code as your “experimental apparatus”.
we go back to the Observe step to gather the results of running the experiment.

2021-11-05

LeetCode - Algorithms - 1668. Maximum Repeating Substring

Problem

1668. Maximum Repeating Substring

Java

Brute-force loop

class Solution {
    private int countByPrefix(String sequence, String word) {
        int maxCount = 0;
        int tempCount = 0;
        int word_len = word.length();
        for (int i = 0; i <= sequence.length() - word_len; i++) {
            String str = sequence.substring(i, i + word_len);
            if (str.equals(word)) {
                tempCount++;
                i += word_len - 1;
            } else {
                tempCount = 0;
            }
            if (tempCount > maxCount)
                maxCount = tempCount;
        }
        return maxCount;
    }

    private int countBySuffix(String sequence, String word) {
        int maxCount = 0;
        int tempCount = 0;
        int word_len = word.length();
        for (int i = sequence.length(); i > word_len - 1; i--) {
            String str = sequence.substring(i - word_len, i);
            if (str.equals(word)) {
                tempCount++;
                i = i - word_len + 1;
            } else {
                tempCount = 0;
            }
            if (tempCount > maxCount)
                maxCount = tempCount;
        }
        return maxCount;
    }
    
    public int maxRepeating(String sequence, String word) {
        int maxCount_prefix = countByPrefix(sequence, word);
        int maxCount_suffix = countBySuffix(sequence, word);
        return maxCount_prefix > maxCount_suffix ? maxCount_prefix : maxCount_suffix;       
    }
}

Submission Detail

212 / 212 test cases passed.
Runtime: 1 ms, faster than 82.84% of Java online submissions for Maximum Repeating Substring.
Memory Usage: 37.5 MB, less than 50.50% of Java online submissions for Maximum Repeating Substring.

Substring search

class Solution {
    public int maxRepeating(String sequence, String word) {
        int count = 0;
        String pat = word;
        while (sequence.contains(pat)) {
            count++;
            pat += word;
        }
        return count;        
    }
}

Submission Detail

212 / 212 test cases passed.
Runtime: 1 ms, faster than 82.48% of Java online submissions for Maximum Repeating Substring.
Memory Usage: 37.6 MB, less than 50.74% of Java online submissions for Maximum Repeating Substring.

2021-11-02

My English Words List - November - 2021

inquire

verb

to ask for information

We inquired the way to the station.

I called the school to inquire about the application process.

flutter

verb

butterflies fluttering among the flowers

We watched the butterflies fluttering in the garden.

Leaves fluttered to the ground.

The bird was fluttering its wings.

The breeze fluttered the curtains.

Flags fluttered in the wind.

scythe

noun

Scythe

A scythe is an agricultural hand tool for mowing grass or harvesting crops. It is historically used to cut down or reap edible grains, before the process of threshing. The scythe has been largely replaced by horse-drawn and then tractor machinery, but is still used in some areas of Europe and Asia.

vine

noun

Grapes grow upon a vine.
Apples grow upon a tree.
Blueberries grow upon a bush.

inflate

verb

inflate a balloon

We used a pump to inflate the raft.

Rapid economic growth may cause prices to inflate.

Low tire pressure: Stop in a safe place, check tire pressures, and inflate tire(s) if necesssary.

pane

noun

a framed sheet of glass in a window or door

frost on a window pane

Paned window (architecture)

swot

noun

They coined a new term in Chinese: xiao zhen zuotijia, meaning “small-town swot”.

Cramming (education)

Synonyms

bookworm
nerd

vague

adjective

He gave only a vague answer.

When I asked him what they talked about, he was rather vague.

I think I have a vague understanding of how it works.

His conjecture was more vague.

marvelous

adjective

Andrew Wiles’s marvelous proof

vacant

adjective

a vacant seat on a bus

a vacant room

These lockers are all vacant.

a vacant job position

leash

noun

put a dog on a leash

Dogs must be kept on a leash while in the park.

Leash

cartoon

noun

The kids are watching cartoons.

Cartoon

barley

noun

Barley

croak

verb

We could hear the frogs croaking by the pond.

blend

verb

The music blends traditional and modern melodies.

noun

a blend of cream and eggs

damp

noun

slight wetness in the air

The cold and damp made me shiver.

verb

His hands were damped with sweat.

Adjective

My hair’s still damp from the rain.

thorn

noun

Thorns, spines, and prickles

wellness

noun

lifestyles that promote wellness

Daily exercise is proven to promote wellness.

Synonyms

fitness, health, healthiness, robustness, wholeness

Antonyms

illness, sickness, unhealthiness

assure

verb

I assure you that we can do it.

I can assure you that you won’t be disappointed.

Hard work assures success.

He assured the children all was well.

She assured herself that the doors were locked.

the seller assured the buyer of his honesty

puck

noun

a vulcanized rubber disk used in ice hockey
a rubber disk used in hockey

Hockey puck

rink

The hockey team is practicing at the rink.

Ice rink

Ice hockey rink

seminar

noun

a seminar bringing together the world’s leading epidemiologists

Seminar

reap

verb

He reaped large profits from his investments.

You’ll reap the benefit of your hard work.

reap a crop

Synonyms

gather, harvest, pick

excerpt

noun

Her latest novel is called “The Air We Breathe” and you can read an excerpt on our website.

verb

This article is excerpted from the full report.

sundae

noun

tailored

adjective

pants bought off the rack never fit me so I have to buy tailored ones instead

growl

verb

his stomach growled

the dog growled at the stranger

I could hear a dog growling behind me.

My stomach’s been growling all morning.

gulp

verb

gulp down a sob

gulp down knowledge

She told him not to gulp his food.

The exhausted racers lay on the ground, gulping air.

threshold

noun

on the threshold of a new age

Percolation threshold

Threshold knowledge

pore

noun

Pore (bread)

porous

adjective

full of small holes

porous wood

capable of absorbing liquids

porous paper

dynamite

noun

Dynamite was invented by Swedish chemist Alfred Nobel in the 1860s and was the first safely manageable explosive stronger than black powder.

Dynamite

verb

They plan to dynamite the old building.

willful

adjective

a stubborn and willful child

willful children

He has shown a willful disregard for other people’s feelings.

The Willful Child

Once upon a time there was a child who was wilful and would not do what her mother wished.

2021-11-01

Be the Best of Whatever You Are

by Douglas Malloch

If you can’t be a pine on the top of the hill,
Be a scrub in the valley — but be
The best little scrub by the side of the rill;
Be a bush if you can’t be a tree.

If you can’t be a bush be a bit of the grass,
And some highway happier make;
If you can’t be a muskie then just be a bass —
But the liveliest bass in the lake!

We can’t all be captains, we’ve got to be crew,
There’s something for all of us here,
There’s big work to do, and there’s lesser to do,
And the task you must do is the near.

If you can’t be a highway then just be a trail,
If you can’t be the sun be a star;
It isn’t by size that you win or you fail —
Be the best of whatever you are!

2021-10-31

Fall, leaves, fall

by Emily Brontë

Fall, leaves, fall; die, flowers, away;
Lengthen night and shorten day;
Every leaf speaks bliss to me
Fluttering from the autumn tree.
I shall smile when wreaths of snow
Blossom where the rose should grow;
I shall sing when night’s decay
Ushers in a drearier day.

2021-10-30

Plane Division by Lines

Problem

How many slices of pizza can a person obtain by making n straight cuts with a pizza knife?

What is the maximum number $ L_n $ of regions defined by n lines in the plane?

Solution

The nth line (for n > 0) increases the number of regions by k if and only if it splits k of the old regions, and it splits k old regions if and only if it hits the previous lines in k-1 different places. Two lines can intersect in at most one point. Therefore the new line can intersect the n-1 old lines in at most n-1 different points, and we must have $ k \leq n $. We have established the upper bound
$ L_n \leq L_{n-1} + n, \qquad \text{for n>0. } $

Furthermore it’s easy to show by induction that we can achieve equality in this formula. We simply place the nth line in such a way that it’s not parallel to any of the others (hence it intersects them all), and such that it doesn’t go through any of the existing intersection points (hence it intersects them all in different places).

recurrence relation

$ L_0=1; $
$ L_n=L_{n-1}+n, \qquad \text{for n>0. } $

Simply put, each number equals a triangular number plus 1.

In other words, $ L_n $ is one more than the sum $ S_n $ of the first n positive integers.

$ L_n = \frac{1}{2}n(n+1)+1 $