Find the nth term of a Quadratic Sequence (Maths GCSE)

Find the nth term of a Quadratic Sequence (Maths GCSE)

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:

nth-term-05

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:

nth-term-04

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 3nfrom 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:

nth-term-03

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 2nfrom 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):

nth-term-06

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 2n+ 4n. Finally we subtract (2n+ 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
2n+ 4n 6 16 30 48
start minus (2n+ 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 2n+ 4n, and that will be our solution: 2n+ 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

nth-term-01

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:

nth-term-07

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:

nth-term-06

If the second layer of gaps is constant, it is a quadratic sequence:

nth-term-03

Basic Rules of Modern Greek – Some, a, the (Cases, definite articles, indefinite articles)

Basic Rules of Modern Greek – Some, a, the (Cases, definite articles, indefinite articles)

I’ve been learning Greek!

This is one in a series of cheatsheets. Full list here.

THE (NOMINATIVE / SUBJECT)

Singular masculine ο ο άντρας = the man
Plural masculine οι οι άντρες = the men
Singular feminine η η γυναίκα = the woman
Plural feminine οι οι γυναίκες = the women
Singular neuter το το παιδί = the child
Plural neuter τα τα παιδιά = the children

 

CASES

Nominative The Subject of the sentence She
Genitive Possessive His
Accusative The Object of the sentence Him
Vocative Calling someone Calling someone

 

A/AN/ONE

MASCULINE FEMININE NEUTER
NOMINATIVE ένας μία or μια ένα
GENITIVE ενός μίας or μιας ενός
ACCUSATIVE ένα or έναν μία or μια ένα

 

THE – CASES

MASCULINE FEMININE NEUTER
Nominative singular ο άντρας = the man η γυναίκα = the woman το παιδί = the child
Genitive singular του άντρα = of the man της γυναίκας = of the woman του παιδιού = of the child
Accusative singular τον άντρα = the man τη γυναίκα = the woman το παιδί = the child
Vocative singular άντρα = man γυναίκα = woman παιδί = child
Nominative Plural οι άντρες = the men οι γυναίκες = the women τα παιδιά = the children
Genitive plural των αντρών = of the men των γυναικών = of the women των παιδιών= of the children
Accusative Plural τους άντρες = the men τις γυναίκες = the women τα παιδιά = the children
Vocative Plural άντρες=men γυναίκες=women παιδιά = children

 

SOME

MASCULINE FEMININE NEUTER
NOMINATIVE μερικοί μερικές μερικά
GENITIVE μερικών μερικών μερικών
ACCUSATIVE μερικούς μερικές μερικά
VOCATIVE μερικοί μερικές μερικά

 

Basic Rules of Modern Greek – Phrases

Basic Rules of Modern Greek – Phrases

I’ve been learning Greek!

This is one in a series of cheatsheets. Full list here.

PHRASES

Καλημέρα Good Morning
Καληνύχτα / Καλό βράδυ Good night
Καλησπέρα Good evening
Όχι No
Ναι Yes
Ευχαριστώ Thanks / Thank you
Παρακαλώ Please / You are welcome
Λυπάμαι I am sorry
Συγνώμη Sorry / Excuse me
Αντίο Goodbye
Σ’ αγαπώ / Σε αγαπώ I love you
Γεια Hi / Hello
Τι κάνεις; How are you? / What are you doing?
Πόσο κάνει; / Πόσο κοστίζει; How much does it cost?
Εγώ είμαι ο / η ….. I am ….
Εγώ ζω (or μένω) στον / στην / στο …. I live in …..

 

Basic Rules of Modern Greek – I (you, she, etc), My (your, her), am, have (Pronouns and auxiliary verbs, conjugation)

Basic Rules of Modern Greek – I (you, she, etc), My (your, her), am, have (Pronouns and auxiliary verbs, conjugation)

I’ve been learning Greek!

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!)

Basic Rules of Modern Greek – The Alphabet

Basic Rules of Modern Greek – The Alphabet

I’ve been learning Greek!

This is one in a series of cheatsheets. Full list here.

Alphabet

Α-α Άλφα Alpha A as in Ant
Β-β Βήτα Veeta V as in Vase
Γ-γ Γάμμα (Γάμα) Gama g as in Good, or y as in Yellow
Δ-δ Δέλτα Delta TH as in THe
Ε-ε Εψίλον Epsilon E as in Element
Ζ-ζ Ζήτα Ζeeta Z as in Zoo
Η-η* Ήτα Eeta EE as in sEE
Θ-θ Θήτα theta th as in Thing
Ι-ι* Ιώτα (γιώτα) Iota EE as in sEE
Κ-κ Κάππα (κάπα) Kapa K as in Kitten
Λ-λ Λάμδα Lambda L as in Lemon
Μ-μ Μυ (μι) Mee M as in Mother
Ν-ν Νυ (Νι) Nee N as in North
Ξ-ξ Ξει (Ξι) Ksee X as in foX
Ο-ο* Όμικρον Omicron O as in Organ
Π-π Πει (Πι) Pee P as in Pet
Ρ-ρ Ρω (ρο) Row R as in Rhapsody
Σ-σ/ς* Σίγμα Sigma S as in Sit
Τ-τ Ταυ Taf T as in Table
Υ-υ* Ύψιλον Ypsilon EE as in sEE
Φ-φ Φει (φι) Fee F as in Fun
Χ-χ Χει (Χι) Chee / Hee H as in Hurry
Ψ-ψ ψει (ψι) Psee PS as in liPStick
Ω-ω* Ωμέγα Omega O as in Organ

 

Η-η, Ι-ι and Υ-υ have the same pronunciation (“ee”)

Ο-ο and Ω-ω have the same pronunciation (“o”)

Sigma has 2 types in lower case: Start of or inside word = σ, but end of word = ς

 

Diphthongs

ΑΙ αι sounds like E-ε, or “eh” as in element
ΕΙ ει sounds like Η-η, Ι-ι, Υ-υ or like ee
ΟΙ οι sounds like Η-η, Ι-ι, Υ-υ or like ee
ΥΙ υι sounds like Η-η, Ι-ι, Υ-υ or like ee
ΑΥ αυ sounds like “av” or “af”
ΕΥ ευ sounds like “ev” or “ef”
ΟΥ ου sounds like “u” as in “soup” .

 

Double consonants

ΜΠ μπ sounds like b
ΝΤ ντ sounds like d
ΓΚ γκ sounds like g
ΓΓ γγ sounds like ng
ΤΣ τσ sounds like ts
ΤΖ τζ sounds like tz

 

Accents

Modern Greek has only ONE accent.

It is placed above the accented vowels, like this: ά, έ, ή, ί, ό, ύ, ώ.

The accent goes on one of the three last syllables.

Accents help you give emphasis to the right syllable.

E.g. “βιβλίο” (veevLEEo), ”μιλώ” (meeLO) etc.

 

Punctuation Marks

The Period, or full stop, the comma and the exclamation mark are the same as English.

The Greek question mark looks just like the English semi colon ;

Basic Rules of Modern Greek

Basic Rules of Modern Greek

I’ve been learning Greek!

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:

The Alphabet

Some, a, the (Cases, definite articles, indefinite articles)

I (you, she etc) am, have (auxiliary verbs and pronouns, conjugation)

Phrases

Don’t confuse the genitive definite article with the possessive pronoun

How to Tie Your Shoelaces

How to Tie Your Shoelaces

! Stop the press !

Since writing this post I have been given an alternative method of tying your shoelaces – with two different demonstrations, both of which appear easier to teach, easier to learn and more effective. This leaves me slightly conflicted, because I’m rather proud of my lovely diagrams. So I’m going to leave the rest of this post in place, despite the fact that the two videos below effectively render it redundant. Oh well.

Original post:

I just dug this out again, for my 9-yr-old. Yeah, he can’t quite tie a shoelace yet. These days you can easily get away with it, because apart from walking boots, most kids’ shoes don’t have laces.

I enjoy teaching, but there is this thing called “Expert-Induced Amnesia“. It’s about that skill you’ve had for so long, you don’t know how you do it. When you try to teach it to other people, you struggle. As a parent, I’ve been starkly reminded of this in two examples: One is riding a bike, and the other is tying your shoelaces.

Teaching a child to ride a bike is extra hard, because you know at some point you’ll have to let go and hope for the best. And have the band-aid ready. But the shoelaces thing… maybe it’s just me, but wow, I found it hard to explain what I do and how I do it. My fingers just know. I don’t think about it. It’s muscle memory. And as soon as I try to slow it down and explain it, I can’t even remember what to do. I can only do it quickly, in a blur.

Also, learners need to practice repeatedly to get it right. Explaining this repeatedly can get wearing, particularly as most of the teaching opportunities come when you’re in a hurry to get out the door, and really you just want the shoes on the feet, with minimal fuss.

One of the hardest parts of teaching is patience. Resisting the urge to do it for them. “Oh, I’ll show you,” you say. You think you’re being helpful, but really you’re just being impatient. They need to do it for themselves.

That was a ridiculously long preamble.

Tl;dr: I forced myself to sit down and work it out, step by step. Then I drew diagrams and stuck them to a piece of card with laces attached. Then I gave it to my eldest son and left him to practice on his own. And now it’s my youngest son’s turn, so I dug the card out again (which is why it looks all tatty and old).

And here it is. Apologies for the tattiness. But just in case you’re teaching somebody to tie shoelaces, or learning to do it yourself… here are some diagrams that just might help.

(Also, apologies if you thought this was going to culminate in some fantastic metaphor, where the laces represent the meaning of life, the universe and everything. It really is a post about shoelaces.)

IMG_5445

IMG_5447

IMG_5448