quadratic equations (white flag goes up)

[citoyen]

I didn't quite understand the question I'm afraid. 

Nor did several native English speakers!

d.

[gonzo]

You did.  reciprocal is so much shorter than to the power of minus one.
Just jargon really, like so many other fields.

But add to that the time taken to explain what the reciprocal is and it's far longer!

Besides, everyone I know has always called it the inverse.

[mattc]

You did.  reciprocal is so much shorter than to the power of minus one.
Just jargon really, like so many other fields.

But add to that the time taken to explain what the reciprocal is and it's far longer!

Besides, everyone I know has always called it the inverse.

I'm pretty sure we were taught 'reciprocal' (but at some point I think I switched to 'inverse' - possibly on my degree?). I'm a couple of school generations older than Gonzo.

I would guess that reciprocal is a more familiar everyday word than inverse to 'normal' people (i.e. before they study any maths!)
-------------

p.s. I've never heard of this 2nd difference stuff. I could solve the problem, and I know about arithmetic and geometric series etc (Proof by recursion anyone? A simple idea that took me ages to get my head round).
I think these things are a way of introducing quadratics without the world of graphs, equations of a straight line, crossing the x-axis etc. I think.

[clarion]

I used to love series.  I remember in one of my maths O level papers, I couldn't remember whether the differential of cos was sin or -sin (or the other way round), so I derived the series and differentiated that

I have not the first idea now how I would go about deriving the series for cosine, let alone in exam conditions and as a piece of rough working out.

I got an A, in case you're worrying

[David Martin]

I think I will have to re-learn English when I do homework with G. I didn't quite understand the question I'm afraid. 

You were not the only one. It was some of the unfamiliar jargon that was used that would have confused me.

..d

[David Martin]

You did.  reciprocal is so much shorter than to the power of minus one.
Just jargon really, like so many other fields.

But add to that the time taken to explain what the reciprocal is and it's far longer!

Besides, everyone I know has always called it the inverse.

But inverse is a broader term than reciprocal. cf. average and mean.

What is the inverse of a matrix? is it a reciprocal?

As to 'the time taken to explain' that only occurs if the person you are talking to hasn't already labelled the concept with that particular piece of jargon.

..d

[citoyen]

I used to love series.  I remember in one of my maths O level papers, I couldn't remember whether the differential of cos was sin or -sin (or the other way round), so I derived the series and differentiated that

I have not the first idea now how I would go about deriving the series for cosine, let alone in exam conditions and as a piece of rough working out.

I got an A, in case you're worrying

I got an A for GCSE maths and I don't think I ever learned what cos and sin were. I knew you had to use them for certain algebraic functions, but only got as far as learning when to press the appropriate button on the calculator. I think I missed a large chunk of the maths syllabus because I moved schools during the second year (what they call year 8 these days) and my new school was somewhat further ahead in maths. Cos and sin were probably covered in the bit I missed.

d.

[Wowbagger]

Should old Harry
Catch a herring
Trawling off America?

[mattc]

I used to love series.  I remember in one of my maths O level papers, I couldn't remember whether the differential of cos was sin or -sin (or the other way round), so I derived the series and differentiated that

You could have worked out which to use by just drawing the 2 curves. Much quicker!

(I quite like maths when different concepts overlap like that, and when there is a visual/geometric answer for something difficult. )

[clarion]

Never heard that, but I remembered anyway.

ftr, I think I was in the last year not to be allowed calculators in maths A levels (1983).  When I went to University to study Engineering, we had to submit our calculators for individual approval.  Mine caused a stir because it was neither an approved nor disapproved model, being bought in Japan.  It was OKd eventually, but I was too crap to pass the exams anyway

Calculators are used too readily.  Kids don't tend to understand what numbers mean.  And, because it can't be keyed into a calculator easily, they tend to struggle with algebra, geometry, set theory and matrices (I confess, these last two I found tricky), let alone calculus, which, once you have the key, you unlock a whole new world. </soapbox>

Edit: mattc - perhaps I could, but I don't tend to work visually, and it was much more satisfying doing it my way

[border-rider]

I got an A for GCSE maths and I don't think I ever learned what cos and sin were

They're the real and imaginary parts of e^iθ

If you know that the nth term of the Taylor Series for e^x is xⁿ/n! you can derive the power series for sin(θ) and cos(θ) from that.  If Clarion did that at O-level, he deserved an A

edit: that may seem supremely obscure, but it's something I use regularly now.  It's a fundamental tool in both RF/radio propagation and AC circuit theory.

[border-rider]

ftr, I think I was in the last year not to be allowed calculators in maths A levels (1983).

We were allowed them from the start of the O-level syllabus onwards.  That would have been about 1976 or 77.

[chris]

ftr, I think I was in the last year not to be allowed calculators in maths A levels (1983).

We were allowed them from the start of the O-level syllabus onwards.  That would have been about 1976 or 77.

I took 'O' level maths in 1980. We had a choice of whether to use a calculator or log tables / slide rule. I chose the later.

[clarion]

ftr, I think I was in the last year not to be allowed calculators in maths A levels (1983).

We were allowed them from the start of the O-level syllabus onwards.  That would have been about 1976 or 77.

You must have had an indulgent exam board.  I did AEB.  We used calculators (sparingly) in class, but there was no chance of having them in exams.

And, er..

If you know that the nth term of the Taylor Series for e^x is xⁿ/n! you can derive the power series for sin(θ) and cos(θ) from that.  If Clarion did that at O-level, he deserved an A

I did, and thanks

I went on to get three maths A levels and another in Physics, which is broadly the same thing.

[Greenbank]

I went on to get three maths A levels and another in Physics, which is broadly the same thing.

(Click for original source...)

[citoyen]

Where do logicians fit in that scheme?

[Frenchie]

Engineers? (Applied physics and maths)

[border-rider]

I went on to get three maths A levels and another in Physics, which is broadly the same thing.

I'm not sure about that.  I've always been reasonably good at physics, but my maths is not great.  Not surprisingly I found a PhD a lot easier than a first degree...

[clarion]

Funnily enough, given the cartoon, I found that (at A level at least), there's an awful lot of crossover between applied maths & physics.  Questions are slightly different, but the concepts are much the same

[citoyen]

Engineers? (Applied phyics and maths)

There's not much that's "applied" about formal logic - it's pretty darn abstract.

Although I suppose it relates more closely to the "real world" than pure maths.

d.

[Panoramix]

Just noticed this thread. A slightly different, and possibly simpler way of solving the original problem -

Original series -


n=                 1   2   3   4    5    6
Original series    12  26  46  72   104  142
First differences    14  20  26  32    38
Second Differences     6   6   6    6


This can be extended to the left to get the 0th member of the series -


n=                 0   1   2   3   4    5    6
Original series    4   12  26  46  72   104  142
First differences    8   14  20  26  32    38
Second Differences     6   6   6   6    6


and the value of n and n₀ substituted into our polynomial -

0n² + 0n + c = 4

or

c = 4

We can now consider the instance where n=1 -

an² + bn + c = 12

ax1² + bx1 + c = 12

or

a + b + 4 = 12

which simplifies to

a + b = 8 or b = 8 - a and a = 8 - b

and the instance where n = 2

an² + bn + c = 25

or

4xa + 2xb + 4 = 26

which simplifies to -

2a + b = 11 or b = 11 - 2a and a = (11 - b) / 2

From the above we have b = 8 - a = 1 - 2a and a = 8 - b = (11 - b) / 2

and solving for a gives a = 3 and for b gives b = 5

Therefore the equation we are looking for is -

3n² + 5n + 4

 I would have solved it in a similar manner. Despite having had my early twenties taken away by the French education system to get maths and physics stuffed in my head, I had never heard of Greenbank's solution! I find it interesting to see the different approaches to a similar problem! 

[Greenbank]

I would have solved it in a similar manner. Despite having had my early twenties taken away by the French education system to get maths and physics stuffed in my head, I had never heard of Greenbank's solution! I find it interesting to see the different approaches to a similar problem! 

Indeed. My solution was designed to highlight a couple of extra things along the way, i.e. the composition of the terms, and working from their composition towards a general formula, and the use of triangular numbers representing accumulation of the second differences. I just wanted to be a bit different...

Without trying to explain things, and just looking for the answer, my solution would have been similar to yours (calculate the term for n=0 which gives c, then use the next two terms to generate two simultaneous equations with two unknowns (a and b) and then solve.)

There is no wrong way of working it out as long as the working is shown and the logic at each step is correct and obvious or described if not. As with all questions like this, the bulk of the marks are for the working/logic with only 1 mark for getting the final answer right.

[Woofage]

I got an A for GCSE maths and I don't think I ever learned what cos and sin were

They're the real and imaginary parts of e^iθ

Yebbut you aren't told that until A-level .

[border-rider]

Yebbut you aren't told that until A-level .

Indeed

If Clarion did that at O-level, he deserved an A

 

[clarion]

There is no wrong way of working it out as long as the working is shown and the logic at each step is correct and obvious or described if not. As with all questions like this, the bulk of the marks are for the working/logic with only 1 mark for getting the final answer right.

The joy of maths!  Just like with routefinding for cycling, there are lots of different ways of reaching the result, with different things to see & experience along the way.  Like chess, too, in fact, Wow