Math Is Fun Forum

#6

Click here to see the video solution for n = 8. (The case for general n is done in exactly the same way.)

#4

#2

Click here for the video solution.

For 16, what do you think the scale factor was?

For 17, how can you apply what you did in 16? (Also, see 18-20 for a clue.)

For 18-20, is there a common scale factor for all three vertices?

Use partial fractions. The integrand can be written as a sum of multiples of 1/x^6, 1/x^4, 1/x^2 and 1/(x^2 + 1). These can then of course be integrated term by term.

The imaginary unit is defined as i² = -1.

(-i)*(i) = -(i*i) = -(i²) = 1.

By convention:

√1 = 1, by convention. However, the equation x² = 1 has two solutions, 1 and -1.

There's an obvious choice of f(x): if you can't see it, try dividing by

The initial condition restricts the choice of f.

Now plug f(x) into your limit, use the fact that log is a continuous function, and use L'Hopital a couple of times to get the answer.

For the first, you could factorise the LHS as (ab + 1)(a + b) = 8(a+b) = 80, i.e. a + b = 10. Then notice that (a + b)^2 = a^2 + 2ab + b^2.

For the second, there probably is some neat way of doing this, but you can do this easily with matrices (write in the form Ax = b, where A is a 5x5 matrix, x = (a,b,c,d,e), b = (177,154,146,138,165)).

For the third, take the equation p + q + r = 7 and multiply first by p, then q, then r to form 3 separate equations in p,q,r, then add them together. You can then re-arrange for pq + qr + rp.

You mean {not B} => {not A}?

But is that really any better than, say, the Sieve of Eratosthenes?

It wouldn't -- using your notation, x is prime iff (a,b) = (1, x) or (x,1). Even if a or b is 1 modulo 10, that doesn't guarantee that a or b is 1, and for large x, will give you a very large number of possibilities.

A chip pan is a deep pan you use to deep fry things in. It's a portable, home-use version of what you might see employees using in a fried chicken shop to fry chips in (they put them in a metal basket and slowly lower them into the oil). You can fry other things aside from chips with a chip pan.

In the linear algebra courses I've taken, it was common not to use any kind of formatting to distinguish between vectors, matrices, scalars, multilinear forms, etc. it simply became too tedious, and it was obvious from the context what each symbol meant. I personally tend to get more put off by all the decorative notation (especially when they start double-underlining or triple-hatting). As bob bundy said, there isn't any definite rule about this, but when dealing with numerous algebraic objects in the same context, these conventions tend to vanish.

However, if you're using just vectors and scalars, then I think a separate format for vectors would be appropriate.

By 'magnitude-bearing form' I guess you mean a unit vector?