I got confused after using the word “his” to explain the genitive case of the definite article (here).
It’s especially confusing because the possessive pronoun for 3rd person plural (ie “their”), is the same word (“τους”) as the accusative form of the definite article for male plural (here), but NOT the same as the genitive form of the definite article for male plural (which is “των”).
Anyway, just to note that the definite article and the possessive pronoun are often, but not always, the same.
I’m currently helping my 15-yr-old son revise for his maths GCSE, and one topic is “finding the nth term of a quadratic sequence”. I’m an ex high school maths teacher, but I had forgotten how to do this. I couldn’t find decent complex examples on either of my favourite GCSE maths revision sites (Maths Genie and BBC Bitesize), and when you’re doing the more complex examples, a step-by-step guide is really useful.
So I’m placing my notes here in case they’re any use to anyone else.
You’re aiming for a result of an2 + bn + c, but easier examples might have a solution of an2 + b, and even easier ones will just be an2.
Simplest Example (an2):
Find the nth term for the following quadratic sequence: 3, 12, 27, 48, …
First calculate the gaps between the numbers – these are 9, 15 and 21.
Then find the gaps between the gaps – these are 6 and 6. Like this:
Take that 6 and divide it by 2 (it’s easy to forget to divide by 2!), to get 3. This tells you that your final result will contain the term 3n2.
I’ve already told you that this is a simple example – we’ve reached our solution: 3n2. But you should always check your results:
n
1
2
3
4
n2
1
4
9
16
3n2
3
12
27
48
Yup, that’s our original sequence.
More Complex Example (an2 + b):
Find the nth term for the following quadratic sequence: 1, 10, 25, 46, …
First calculate the gaps between the numbers – these are 9, 15 and 21.
Then find the gaps between the gaps – these are 6 and 6. Like this:
Take that 6 and divide it by 2 (it’s easy to forget to divide by 2!), to get 3. This tells you that your final result will contain the term 3n2.
Create a grid, which starts with your original sequence. Below that, add whatever rows you need to help you calculate 3n2.
Now, subtract 3n2 from the original sequence. So in the below grid, we subtract the fourth row from the first row, and that gives us a new sequence, which we have placed in the fifth row:
start
1
10
25
46
n
1
2
3
4
n2
1
4
9
16
3n2
3
12
27
48
start minus 3n2
-2
-2
-2
-2
We now have a row of constant numbers. This tells us we can reach a solution. It tells us to add -2 to 3n2, and that will be our solution: 3n2 – 2.
We can easily check this by adding up the fourth and fifth rows, which gives us the first row (the original sequence).
Most Complex Example (an2 + bn + c):
Find the nth term for the following quadratic sequence: -8, 2, 16, 34, …
First calculate the gaps between the numbers – these are 10, 14 and 18.
Then find the gaps between the gaps – these are 4 and 4. Like this:
Take that 4 and divide it by 2 (it’s easy to forget to divide by 2!), to get 2. This tells you that your final result will contain the term 2n2.
Create a grid, which starts with your original sequence. Below that, add whatever rows you need to help you calculate 2n2.
Now, subtract 2n2 from the original sequence. So in the below grid, we subtract the fourth row from the first row, and that gives us a new sequence, which we have placed in the fifth row:
start
-8
2
16
34
n
1
2
3
4
n2
1
4
9
16
2n2
2
8
18
32
start minus 2n2
-10
-6
-2
2
We don’t have a row of constant numbers yet, so we need to keep working. We need to look at the gaps between the numbers in our new sequence (in the bottom row of the table):
Now we have found a constant difference. This tells us that there will be a 4n in our answer. Note that this is because we have found a linear sequence. Note also that in the case of a linear sequence, we do NOT divide the number by 2.
So now we add some more rows to our grid. First we calculate 4n, and then we calculate 2n2 + 4n. Finally we subtract (2n2 + 4n) from our original sequence (subtract the 7th row from the first row):
start
-8
2
16
34
n
1
2
3
4
n2
1
4
9
16
2n2
2
8
18
32
start minus 2n2
-10
-6
-2
2
4n
4
8
12
16
2n2 + 4n
6
16
30
48
start minus (2n2 + 4n)
-14
-14
-14
-14
We now have a row of constant numbers. This tells us we can reach a solution. It tells us to add -14 to 2n2 + 4n, and that will be our solution: 2n2 + 4n – 14.
We can easily check this by adding up the seventh and eighth rows, which gives us the first row (the original sequence).
More worked complex examples
Note that in this next one there is a NEGATIVE difference between the terms of the sequence on row 5. This one can easily catch you out. Rather than thinking of the difference between the numbers, it helps to ask yourself, “how do I get from each term to the next one?” The answer in this case is, “subtract one”. This one can also look a little tricky because it contains fractional numbers, but you just follow the same rules as before:
Telling the Difference Between a Linear Sequence (an + b) and a Quadratic Sequence (an2 + bn + c).
When we calculate gaps between the numbers in the sequence, if the first level of gaps is constant, this means it is a linear sequence:
If the second layer of gaps is constant, it is a quadratic sequence:
This is one in a series of cheatsheets. Full list here.
PRONOUNS
If “they” refers to a group all males or male and female or its gender composition is unknown, αυτοί is used.
Εγώ
I
Εσύ
you (singular)
Εσείς
you (plural)
Εμείς
we
Αυτός
he
αυτή
she
αυτό
it
Αυτοί
they (male)
αυτές
they (female)
αυτά
they (neuter)
TO BE
Important note: the pronoun (Εγώ, εσύ) …is not always needed.
Εγώ είμαι
I am
Εσύ είσαι
you (singular) are
Εσείς είσαστε
you (plural) are (or είστε)
Εμείς είμαστε
we are
αυτή είναι
she is (or he, or it)
αυτές είναι
they (female) are
TO HAVE
Singular
Plural
First Person
I have – έχω (“echo”)
we have – έχουμε, έχομε
Second Person
you have – έχεις
you have – έχετε
Third Person
she has – έχει
they (f) have – έχουν, έχουνε
First Conjugation Verbs
Many Greek verbs fall into this same pattern for changing their endings (or conjugating.)
We call this group of verbs the first conjugation verbs.
Here are a few more of them, given, as always, in the first person form:
I see
βλέπω
I buy
αγοράζω
I drink
πίνω
I know
ξέρω
I take
παίρνω
I give
δίνω
I eat
τρώω
POSSESSIVE PRONOUNS:
Person
Pronoun (own one thing)
Pronoun (own many things)
1st person singular
(Δικός/Δική/Δικό) μου
(Δικοί/Δικές/Δικά) μου
2nd person singular
(Δικός/Δική/Δικό) σου
(Δικοί/Δικές/Δικά) σου
3rd person singular (masculine)
(Δικός/Δική/Δικό) του
(Δικοί/Δικές/Δικά) του
3rd person singular (feminine)
(Δικός/Δική/Δικό) της
(Δικοί/Δικές/Δικά) της
3rd person singular (neuter)
(Δικός/Δική/Δικό) του
(Δικοί/Δικές/Δικά) του
1st person plural
(Δικός/Δική/Δικό) μας
(Δικοί/Δικές/Δικά) μας
2nd person plural
(Δικός/Δική/Δικό) σας
(Δικοί/Δικές/Δικά) σας
3rd person plural
(Δικός/Δική/Δικό) τους
(Δικοί/Δικές/Δικά) τους
(masc/fem/neuter)
(masc/fem/neuter)
EXAMPLES:
Ο άντρας μου=My husband
Ο δικός μου άντρας= My own husband (emphatic).
WHAT’S THE DIFFERENCE BETWEEN Δικός, δική, δικό?
Δικός is used if the owned object is of masculine gender: Ο άντρας είναι δικός μου=The man is mine.
Δικός becomes δικοί when the owned object of masculine gender is in plural.
So, οι άντρες είναι δικοί μου=the men are mine.
Δική is used if the owned object is of feminine gender: Η γυναίκα είναι δική μου=The woman is mine.
Δική becomes δικές when the owned object of feminine gender is in plural.
So, οι γυναίκες είναι δικές μου=the women are mine.
Δικό is used if the owned object is of neuter gender: Το παιδί είναι δικό μου=The kid is mine.
Δικό becomes δικά when the owned object of neuter gender is in plural.
So, τα παιδιά είναι δικά μου=the children are mine.
THE DOUBLE ACCENT RULE
When μου,σου,του,της,μας,σας,τους comes after a word that is accented on the antepenult (second syllable from the end e.g. αυτοκίνητο), then it is accented also on the last syllable.
Example:
το αυτοκίνητό μου=my car
το ραδιόφωνό της= her radio
η τσάντα του=his bag (no double accent here because the word τσάντα is not accented on the antepenult!)
I’ve been using DuoLingo, which is great in some ways, but utterly bewildering in others. The app asks you to remember random sentences and words with no apparent attempt to explain any basic grammatical rules or word endings. In fact that info is available on the DuoLingo website, but is still a bit haphazard even there, so I’ve created some cheat sheets with some useful basic rules:
