Alternate proof to induction for integer sum | Precalculus | Khan Academy
A1 A2 B1 B2 C1 C2
what
US / wɒt /
•UK / hwɑt, hwʌt, wɔt, wʌt,hwət, wət /
- pronoun
- Thing or things that; the same kind as
- Used to emphasize something you are about to say
- adjecitve
- Used to refer to a particular amount, or number
- Used to place emphasis on a particular quality
- adverb
- To a certain degree
- determiner
- Question words used for asking for information
- exclamation
- Used to express excitement, shock, or surprise
- Used to ask people to repeat something they said
get
US / ɡet /
•UK / ɡɛt /
- verb
- To become affected by illness or disease
- To begin to feel or understand an emotion or idea
- To send or bring someone or something home, etc.
- To go somewhere to obtain something
- To (cause to) do a particular thing
- To obtain, receive or be given something
- To currently have
- To prepare for use
- To understand something being said or read
that
US / ðæt /
•UK / ðæt, ðət /
- adjecitve
- The person or thing being talked about
- Used to identify something both the speakers know
- adverb
- Used to reinforce adjectives and adverbs
- To such a degree
- conjunction
- Used to connect a noun clause
- Used to introduce an adverbial clause
- Used to introduce a noun clause
- determiner
- Used as a noun to refer to something
- pronoun
- Referring an object far away
- Used to refer to the relative pronoun ‘which’
each
US / i:tʃ /
•UK / itʃ /
- determiner
- Every one of two or more things
- pronoun
- Per one person or thing
different
US / ˈdɪfrənt /
•UK / ˈdɪfrənt /
- adjecitve
- Not of the same kind; unlike other things
interesting
US / ˈɪntrəstɪŋ /
•UK / ˈɪntrɪstɪŋ, -tərɪstɪŋ, -təˌrɛstɪŋ /
- adjecitve
- Taking your attention; making you want to know
- verb
- To make someone want to know about something
- To persuade to do, become involved with something
twice
US / twaɪs /
•UK / twaɪs /
- adverb
- Two times
then
US / ðen /
•UK / ðɛn /
- adverb
- A time that is not now
- Following another thing in time or arrangement
- Following on reasonably from what was just stated
- At that time, not now
include
US / ɪnˈklu:d /
•UK / ɪnˈklud /
- verb
- To make someone, something part of a group
right
US / raɪt /
•UK / raɪt /
- adjecitve
- Correct or true
- The south side of you looking at the rising sun
- adverb
- Exactly in a place or time
- Being appropriate for a particular event
- In a straight or direct manner
- noun
- Something you are legally or morally allowed to do
- Ideal of what is just and good
- Political view based on conservatism
- verb
- To fix something previously wrong or unjust
- To make upright, as a boat that has turned over
down
US / daʊn /
•UK / daʊn /
- adjecitve
- Being sad, sick, or unhappy
- Not operating, working, or functioning as expected
- adverb
- In a sitting or lying position
- Towards or on the ground, floor, or bottom
- Written or copied onto a piece of paper or book
- noun
- A period of play in American Football
- The soft, smooth hairs that cover a bird's body
- preposition
- Along something's path, course, or direction
- Going from a higher position to a lower position
- verb
- To eat or drink very quickly and without stopping
again
US / ə'ɡen /
•UK / əˈɡɛn /
- adverb
- One more time, once more
finally
US / ˈfaɪnəli /
•UK / ˈfaɪnəli /
- adverb
- used especially at the beginning of a sentence to introduce the last point or idea
- After a long time or some difficulty
until
US / ənˈtɪl /
•UK / ʌnˈtɪl /
- conjunction
- Up to; to a time when something happens
- preposition
- Before (a specific time)
something
US / ˈsʌmθɪŋ /
•UK / ˈsʌmθɪŋ /
- adverb
- In some degree or measure
- noun
- a person or thing of some value or consequence
- pronoun
- Thing that is not yet known or named
- Approximately
any
US / 'enɪ /
•UK / ˈɛni /
- adjecitve
- One, or some, no matter which
- pronoun
- One (thing) of many; some
only
US / ˈəunli /
•UK / ˈonli /
- adverb
- Just one; just that amount or thing; not more than
- Not very important or serious
- Just that type of thing, person
- Short time ago; not before a certain time
- conjunction
- Except; but
minus
US / ˈmaɪnəs /
•UK / ˈmaɪnəs /
- adjecitve
- Being close to the bottom of a letter grade
- noun
- Negative quality; disadvantage
- Sign that indicates subtraction
- preposition
- Being less than zero in number
- Used when subtracting one number from another
- verb
- To take one number away from another; subtract
leave
US / li:v /
•UK / liv /
- noun
- Permission to do something
- Vacation time; time off work
- verb
- To go away from; depart
- To gift property to someone after you die
- To allow something to continue happening
- To let something remain unused or available
- To let something or someone remain somewhere
over
US / 'əʊvə(r) /
•UK / ˈovɚ /
- adverb
- Finished; no longer happening
- Towards the ground/away from a standing position
- noun
- A set of six consecutive balls bowled in cricket
- other
- Finished
- preposition
- Showing the relation when a thing is above another
- On or covering something
- Above or across something
- More than you need
- From one side to the other
- In the direction of; toward
integer
US / ˈɪntɪdʒə(r) /
•UK / ˈɪntɪdʒɚ /
- noun
- Math a whole number
two
US / tu: /
•UK / tu /
- number
- 2
where
US / hwɛə /
•UK / hwɛr, wɛr /
- conjunction
- At or in the place which
- In whatever place
- pronoun
- Word you use to ask the location of something
proof
US / pru:f /
•UK / pruf /
- noun
- Act of showing that something exists or is true
- Version of a document to be checked for errors
- Special coating that protects something
- Something which shows that something else is true
- verb
- To read a written work to check for mistakes
- To protect by painting on a special coating
and
US / ənd /
•UK / ənd, ən,ænd /
- conjunction
- Used to refer to two or more things
- Plus; in addition; on top of that
- Used to introduce an action that follows another
try
US / traɪ /
•UK / traɪ /
- noun
- Act of putting the ball over the rugby goal line
- verb
- To make an effort, to attempt to do something
- To examine someone or something in a court of law
- To test to see if it is good or you like it
- To make a person go through hardship; test
- To examine something to see if it works
will
US / wɪl /
•UK / wɪl /
- noun
- Desire or choice of someone
- Document saying who gets your money when you die
- Strong desire to do something; determination
- Desire to do something; strong wish to do
- other
- Used with verbs to express the future
- verb
- To leave property to someone after your death
- To use mental effort to make something happen
- Am (is, are) likely to do
- To influence someone to do something
- To make something happen by strongly wanting it
exactly
US / ɪgˈzæktli /
•UK / ɪɡˈzæktli /
- adverb
- No more and no less than; precisely
here
US / hɪə(r) /
•UK / hɪr /
- adverb
- In this place; at this location, position or time
once
US / wʌns /
•UK / wʌns /
- adverb
- One time; one instance
- At a time in the past
- conjunction
- When; as soon as
side
US / saɪd /
•UK / saɪd /
- adjecitve
- In addition to the central or most important thing
- Next to, or very close to, the right or left of
- noun
- Position or opinion that is opposite to another
- Right or left part of a person's body
- Feature of someone's personality
- Straight edge of an object
- One of two or more groups fighting each other
- verb
- To decide to agree with one point, not the other
divide
US / dɪ'vaɪd /
•UK / dɪˈvaɪd /
- verb
- To split numbers by another number, e.g. 6 / 2 = 3
- To separate something into equal pieces
itself
US / ɪtˈself /
•UK / ɪtˈsɛlf /
- pronoun
- Referring to it
many
US / 'menɪ /
•UK / ˈmɛni /
- determiner
- Used to refer a large number of things
use
US / ju:s /
•UK / ju:s /
- noun
- Ability to employ something
- Need to enable it to work
- Permission to do something with something
- Act of spending money
- Operation of a machine for its function
- verb
- To have; to eat, drink, consume etc.
- To need to operate or work
- To operate a machine, e.g. a computer
- To treat a person badly
- To do something with, for a task or purpose
actually
US / ˈæktʃuəli /
•UK / ˈæktʃuəli /
- adverb
- Used to add new (often different) information
plus
US / plʌs /
•UK / plʌs /
- noun
- A benefit or advantage
- preposition
- In addition; add
- In addition to; Indicates a number above
- Having a value greater than zero
equivalent
US / ɪˈkwɪvələnt /
•UK / ɪˈkwɪvələnt /
- adjecitve
- Equal to something in value, use or meaning
- noun
- Thing like another in quality, quantity or degree
write
US / raɪt /
•UK / raɪt /
- verb
- To compose letters and words on paper or a screen
- To invent or create a computer program
- To create a song or piece of music
pure
US / pjʊə(r) /
•UK / pjʊr /
- adjecitve
- Not mixed with anything else
- Being morally good
last
US / lɑ:st /
•UK / læst /
- adjecitve
- Most recent; the latest
- The one before
- adverb
- Final; with nothing following; after all others
- Most recent; the latest
- determiner
- The one at the end
- verb
- To remain or continue for a certain period of time
exist
US / ɪɡ'zɪst /
•UK / ɪɡˈzɪst /
- verb
- To be present, alive or real
have
US / həv /
•UK / hæv /
- verb
- To drink, smoke, eat or use something
- To experience the effects of something
- To organize an event, or join in an activity
- To own, possess, or hold something
- To cause to happen or produce a particular effect
know
US / nəʊ /
•UK / noʊ /
- verb
- To be familiar with a person or place, thing
- To feel sure or be convinced about something
- To discover or be aware of something
- To be able to distinguish
- To have knowledge of things
count
US / kaʊnt /
•UK / kaʊnt /
- noun
- Number of things added together, e.g. votes
- With the number of crimes, knockouts, as stated
- verb
- To add things together to find the total number
- To matter or be important
- To include someone or something in a final number
could
US / kəd /
•UK / kʊd /
- other
- Past form of 'can' to mean have ability
- Used to ask for something politely, or to offer
rewrite
US / ˌri:'raɪt /
•UK / riˈraɪt /
- verb
- To write again to improve, or correct errors
all
US / ɔ:l /
•UK / ɔl /
- adverb
- Completely; totally
- determiner
- Being every one of something
- pronoun
- 100% of something; the complete amount
expression
US / ɪk'spreʃn /
•UK / ɪkˈsprɛʃən /
- noun
- Act of making your thoughts and feelings known
- Group of words that have a specific meaning
show
US / ʃəʊ /
•UK / ʃo /
- noun
- False display of love, emotion or action
- An event for displaying or promoting goods
- Play, opera or theater production
- Presentation that is broadcast on TV or radio
- verb
- To be easily seen or displayed
- To display your emotions or feelings
- To let someone see something; display
- To explain or teach how something is done
- To lead someone somewhere; point them to something
- To prove something to be true, correct
- To reveal or allow something to be seen
both
US / bəuθ /
•UK / boθ /
- determiner
- Used to refer to two things at the same time
- pronoun
- Used to refer to two things at the same time
they
US / ðeɪ /
•UK / ðe /
- pronoun
- Two or more people, animals, or things
add
US / æd /
•UK / æd /
- verb
- To include as well
- To put two things or numbers together, e.g. 2+2
simple
US / 'sɪmpl /
•UK / ˈsɪmpəl /
- adjecitve
- Not hard to understand or do; not complex
- Having only a few parts; not complex
- Being ordinary or normal
- Being plain; without a lot added for style
prove
US / pru:v /
•UK / pruv /
- verb
- To demonstrate truth by providing evidence
way
US / weɪ /
•UK / we /
- adverb
- Used to emphasize the amount of something
- noun
- Particular aspect of something being thought about
- One of various choices that could be made
- Particular existing condition or state
- (Often long) distance or length of time
- How something is done
- Route to go to a place
- Small street
- Usual behavior, emotions or habits of someone
time
US / taɪm /
•UK / taɪm /
- noun
- Speed at which music is played; tempo
- Point as shown on a clock, e.g. 3 p.m
- Number of hours, minutes needed to do something
- Occasion when something happens
- Period or occasion that something occurred
- Period in history or the past
- Something measured in minutes, hours, days, etc.
- How long an event takes; duration
- verb
- To check speed at which music is performed
- To choose a specific moment to do something
- To measure how long an event takes, e.g. a race
- To schedule something to occur at a specific moment
well
US / wel /
•UK / wɛl /
- adjecitve
- Being in a fortunate, good condition or state
- Healthy and in good condition
- adverb
- In an acceptable, proper, or successful manner
- In a complete, total, or full manner
- A great and considerable degree
- In a successful way, in a good or satisfactory way
- exclamation
- Used to begin or continue a particular statement
- Used to show you want to start saying something
- Used to express that you are waiting for a reply
- You say this to change the topic of a conversation
- Used to indicate you are making a final remark
- noun
- Long deep hole in the ground to get water
- verb
- To cause your eyes to fill with tears
positive
US / ˈpɒzətɪv /
•UK / ˈpɑzɪtɪv /
- adjecitve
- Showing agreement or support for something
- Being sure about something; knowing the truth
- Having the charge produced by electrons
- Being good or useful
- Showing that a chemical is present in a test
just
US / dʒʌst /
•UK / dʒʌst /
- adjecitve
- Proper or appropriate; as is deserved
- Doing or being what is right or fair
- Legally correct; having enough evidence
- adverb
- Almost; nearly
- Exactly at the time of
- Exactly (the same as, like)
express
US / ɪk'spres /
•UK / ɪkˈsprɛs /
- adjecitve
- Sent by a fast mail service
- Traveling at high speed
- noun
- Fast train or bus
- verb
- To send something by fast mail
- To state or show what you think or how you feel
define
US / dɪ'faɪn /
•UK / dɪˈfaɪn /
- verb
- To explain the meaning of words
- To set or mark the limits of something
algebraic
US / ˌældʒɪ'breɪɪk /
•UK / ˌældʒəˈbreɪk /
- adjecitve
- Concerning algebra
for
US / fə(r) /
•UK / fɔr,fə /
- preposition
- Used to show the purpose, or need of something
really
US / ˈri:əli /
•UK / ˈriəˌli, ˈrili /
- adverb
- Used to stress something that is said; definitely
- In a true or accurate manner
- Very
sum
US / sʌm /
•UK / sʌm /
- noun
- Amount when all is added together; total
- verb
- To add two or more numbers together; total
now
US / naʊ /
•UK / naʊ /
- adverb
- At the present time or moment
- Right away or in the next possible moment
- Used to refer to something that just happened
- Time between the present and a moment in the past
- conjunction
- Resulting from or because of something
equal
US / 'i:kwəl /
•UK / ˈikwəl /
- adjecitve
- Same in shape, size, or number
- verb
- To add up or be the same as
pick
US / pɪk /
•UK / pɪk /
- noun
- Sharp tool used for breaking the ground
- Item selected from a group of choices
- verb
- To take fruit or vegetables from a plant or tree
- To decide on a thing from various choices; select
- To remove (flower, etc.) with your fingers
video
US / 'vɪdɪəʊ /
•UK / ˈvɪdiˌo /
- adjecitve
- Of a moving film, with sound
- noun
- Recording capturing action with sound
- verb
- To make a moving film of, with sound
can
US / kæn /
•UK / kən,kæn /
- noun
- A metal container for storing food
- other
- Able to; have the skill to; have the time to
- verb
- To have permission to
- To be possible to do
- To preserve food by sealing it in a metal container
whole
US / həul /
•UK / hol /
- adjecitve
- Complete or full; all of
- Being in a good condition
- noun
- Complete amount or sum; total
definition
US / ˌdefɪˈnɪʃn /
•UK / ˌdɛfəˈnɪʃən /
- noun
- Explanation of word's meaning, as in dictionaries
- The sharpness of an image on a screen
- Visual clearness in a photo or on a screen
would
US / wʊd /
•UK / wʊd /
- other
- Used to indicate something imagined
- Used to ask or request something politely
- Verb used to indicate possibility, request etc.
- Used to indicate a preference
- Used to indicate a reaction
- Indicates something that occurred repeatedly
- Used to indicate willingness or ability to do
not
US / nɒt /
•UK / nɑt /
- adverb
- Word indicating the negative
- (Used to form the negative of verbs)
want
US / wɔnt /
•UK / wɑnt, wɔnt /
- noun
- Something you desire or wish for
- State or condition of poverty
- verb
- To desire or wish for something; hope for a thing
but
US / bət /
•UK / bʌt,bət /
- conjunction
- Used before you say something different, opposite
same
US / seɪm /
•UK / sem /
- adjecitve
- Being equal in quality and amount
- Identical; not different
- Not changing; constant
this
US / ðɪs /
•UK / ðɪs /
- adjecitve
- Used to indicate something already been discussed
- Used to indicate something happening around now
- adverb
- An indication by a physical motion or gesture
- A specific amount of
- determiner
- Person, thing, or idea near you
- pronoun
- Person, thing that has already been discussed
you
US / jʊ /
•UK / ju /
- pronoun
- Person someone is speaking or writing to
- Person or people in general
the
US / ðə /
•UK / ðə /
- article
- Used to refer to something already mentioned
- Used to show there is only one of something
- determiner
- Used to describe all of a family
let
US / let /
•UK / lɛt /
- verb
- To allow someone to do something
- Introducing a suggestion to do something together
- To rent a house, etc. to others
also
US / 'ɔ:lsəʊ /
•UK / ˈɔlso /
- adverb
- In addition; too; in a similar way
kind
US / kaɪnd /
•UK / kaɪnd /
- adjecitve
- In a caring and helpful manner
- friendly and considerate
- noun
- One type of thing
how
US / haʊ /
•UK / haʊ /
- adjecitve
- To what degree? in what amount?
- adverb
- (Used to explain the way to do something)
- exclamation
- For what purpose or reason?
- (Used to ask someone or something's condition)
order
US / 'ɔ:də(r) /
•UK / ˈɔrdɚ /
- noun
- Religious or social group, as of nuns
- Instructions to others to do what you want
- State in which things are neat and organized
- What you ask to buy, e.g. food in restaurant
- Social or political control of an area
- verb
- To put things in a sequence, e.g. a, b, c
- To give instructions to others to do what you want
- To ask for something you want to buy, e.g. a meal
term
US / tɜ:m /
•UK / tɚm /
- noun
- Conditions applying to an agreement, contract
- Length of time something is expected to happen
- Fixed period of weeks for learning at school
- The (precise) name given to something
- verb
- To call; give a name to