G C S E Sequence

[kuiperbelt]

Please see link below:

https://ibb.co/7yMhXsV

[bhg481]

25

[Prudie]

25
But are these questions for our fun and competition or do you really need to know the way of working??

[ClareTG0ld]

Insert n=6 and n=3 separately, and use the answers you're given, to find that

36w + 6b = 42
9 w + 3b = 3

two simultaneous equations, in two unknowns, and can be solved in various ways. The one that feels most natural is to double the second equation and subtract it from the first, and -- separately -- to quadruple the second and subtract from the first, to get the two new equations

18w = 36
-6b=30

from which we get w = 2 and b = -5. Finally, the fifth term in the sequence is given by 5^2*2 + 5*(-5) = 25.

I have skipped a few steps here, but hopefully there is enough that someone can follow it, and fill in the gaps that are left.

[bhg481]

I did it exactly as ClareT - I was just too lazy to type out all the workings.

[kuiperbelt]

My grandson kicked himself when he realised this was a Sim Eqs question. Thank you Clare!!

[ClareTG0ld]

You're welcome!

This is kind of what I mean about how there's a missing step here, where you either (a) instantly realise what's going to be needed here, or (b) mess around with what you *do* know, until you recognise how to use that information to solve things.

It helps, perhaps, to verbalise what's been given.

"OK, so I know that the sequence is wn^2 + bn, but I don't know w and b, so that's to say I don't know TWO things. I do know n, though, it's 3 or 5 or 6. And I am told what the answer is for n=3 or n=6, so that means that I know TWO answers. Oh, hang on, TWO things I don't know, and TWO pieces of info involving them -- that's simultaneous equations!!"

I'm emphasising the "two" a lot here, because that's the "in" to look for. You're looking to match the number of things you don't know with as many pieces of info again -- and, at GCSE level at least, you'll only ever be asked to solve two equations in two unknowns (it's not like solving three/three or more is harder per se, it's just more tedious, and easier to go wrong, and just never comes up).