Math Is Fun Forum

Mainly X-Files.

Yes.

Indeed

Since the sum of angles of a triangle is 180 degrees, or

radians, I would say the answer is

, or rather,

while the sum of the other angles,

goes to 0. However, I'd assume that, if we say that, if the two equal angles approach 0, then we'd simply get a straight line.

Why do you think a sphere can't have tangent lines? At a given point, a point on the surface of a sphere has infinitely many tangent lines, which make up the tangent plane at that point.

An asymptote is a line, so it has the width of a line, which I guess is 0. Since tan(x) has asymptotes at the points

, the second part sounds like you're asking about the distance between the values

for

.

The Fibonacci sequence is defined by F[sub]0[/sub]= 1, F[sub]1[/sub]= 1,  F[sub]n+2[/sub]= F[sub]n+1[/sub]+ F[sub]n[/sub].

   A standard method of trying to solve such recursion formulas is to try something of the form F[sub]n[/sub]= a[sup]n[/sup].  Then, of course, F[sub]n+1[/sub]= a[sup]n+1[/sup] and F[sub]n+2[/sub]= a[sup]n+2[/sup] so the equation becomes a[sup]n+2[/sup]= a[sup]n+1[/sup]+ a[sup]n[/sup].  If we divide the entire equation by a[sup]n[/sup] we arrive at a[sup]2[/sup]= a+ 1 or the quadratic equation a[sup]2[/sup]- a- 1= 0.  Solving that by the quadratic formula,

That tells us that

and

satisfy that equation.  Since this is a linear equation, any solution must be of the form

for some numbers A and B.
In particular, if n= 0

and, if n= 1

  Solving those two equations for A and B, we get

and

Putting all of that together, we have

Well, in my experience, the best teachers can often be the least approachable - but they are best at their job. I've always preferred to work on my own, though.

To find purpose.

Powers are also quite important - I suggest you try to remember these, alongside the powers of 2 in-between;

2[sup]10[/sup] = 1024
2[sup]20[/sup] = 1048576
2[sup]30[/sup] = 1073741824 (not used a lot)

Applications of the above are very vast, but here are some basic examples of where they could be used in basic arithmetic;

32 × 16 = 2[sup]5[/sup] × 2[sup]4[/sup] = 2[sup]9[/sup] = 512
128 × 512 = 2[sup]7[/sup] × 2[sup]9[/sup] = 2[sup]16[/sup] = 65536

Squares are obviously important, too;

1² = 1
2² = 4
3² = 9
4² = 16
5² = 25
6² = 36
7² = 49
8² = 64
9² = 81
10² = 100
11² = 121
12² = 144
13² = 169
14² = 196 (it's just the 69 from 13² reversed - that's how I first remembered it)
15² = 225
16² = 256
17² = 289
18² = 324
19² = 361
20² = 400
25² = 625
30² = 900
40² = 1600
50² = 2500
100² = 10000

Those last ones aren't terribly important, since the answers can be found very easily.

On to cubes - they often saved me a lot of time if the exponent was unknown, i.e. 3[sup]x[/sup] × 4[sup]x[/sup] = 12[sup]x[/sup]

1³ = 1
2³ = 8
3³ = 27
4³ = 64
5³ = 125
6³ = 216
7³ = 343
8³ = 512 (note: 8³ = (2³)³ = 2[sup]9[/sup])
9³ = 729
10³ = 1000
11³ = 1331
12³ = 1728

That's some of what you should try to memorise, alongside the tables bobbym mentioned earlier. Arithmetic is fundamental.

I just quickly skimmed over your post, and I thought it read;

...I need sleep. My concentration levels are drooping a smidge.

... Back to pringles!

Welcome!

Welcome.