A pattern that has two ways you can describe it as!

Take this sequence: 2, 8, 18, 32, 50, 72, 98, 128, 162, 200, 242, 288...

Can you figure out the equations?

-----------------------------------------------------------------------------------------------------------------

Hint: The first way deals with exponents and is a geometric sequence.
The other way is by adding and involves an arithmetic sequence.

-----------------------------------------------------------------------------------------------------------------

Answer:

#1: Take every even number (or any number multiplied by 2) and square it, then divide by 2. You can see it as (2x)². This also equals 4x², which simplifies to 2x². This means you
                                 ------                           -----    could alternatively take every number
                                    2                                2       and multiply by 2. They both get to the
                                                                               same number every time!

#2 Notice how the numbers go up. 2 + _6_ = 8                         8 + _10_ = 18
       18 + _14_ = 32                       32 + _18_ = 50                    50 + _22_ = 72
       72 + _26_ = 98                       98 + _30_ = 128                 128 + _34_ = 162
     162 + _38_ = 200                   200 + _42_ = 242                  242 + _46_ = 288
The first number it goes up by is 6, then that number goes up by 4 every time. This number goes by the arithmetic equation of 4x+2. The only way I can think of to find a number in the sequence without going through the whole thing is to make a new equation using the one I just mentioned. For the 4x, it is (4x1)+(4x2)+(4x3).... which = 4(1+2+3...) To add up all those numbers quickly if you want to do a big number, add the first and the last, the second and the second to last, the third and the third to last... You will notice that all those numbers are the same. Now just take that number and multiply by the number of pairs you have. Now you have to add the twos. Add 2 multiplied by whatever number in the sequence you want, then plus 2 because the starting number is 2, not 0.
So, to get the number in the sequence, all you have to do is multiply 4 by the total of the 1 and 2 and 3 and all that and then add 2x+2 or 2(x+1).

Now, lets try with 15. The number above only go up to 11, so lets do some more. 288+50=338 +54=392 +58=450 +62=512 So now we know that the answer should be 512. So first do 4 multiplied by 1 to 15. To add them up quickly, I'll do 1+14 and 2+13 and 3+12, and so on, until 7+8. All these add up to 15. The seven pairs multiplied by 15 is 105, plus the final term of 15 is 120. 120 times 4 is 480. Then the 15 x 2, +2 = 32. 480+32=512!

One final note: every term that you add on (the one that goes up by 4) should end in 0 every 2nd, 7th, 12th, 17th and so on time that it is added. It should have gone up by 20 every time this happens. Also, Every term that is the total (that goes up quicker and quicker) should end in zero every time the term number ends in 4 or 9. Also, it goes up 100 more than the last time 5 terms ago. The first is 50, so 50 more than last one, which was 0. Then it's 200, 150 more than last. then 450, 250 more. Then 800, 350 more. We can predict 2 of the 5 numbers in each set this way, the 2nd and 4th in it.
The numbers up to 20 (every 5th is bolded): 8, 18, 32, 50, 72, 98, 128, 162, 200, 242, 288, 338, 392, 450, 512, 578, 648, 722, 800, 882

Flame changes ink color!

I noticed another thing today... When I was burning a few scraps of paper with typed letters on them I noticed that in the ashes that remained, if you could still see the words, most of the words had turned orange! I tried it again, as I thought maybe it had been orange in the beginning and I just hadn't noticed... but it still turned orange! When you think of fire, you usually think of orange or yellow and that's the color of the fire from my lighter. I wonder, if I burned the paper with blue flame, would the letters have turned blue...?

Strange Exponent Sequence

This is a strange sequence involving exponents. It is as follows: You pick any positive number. Take the last digit. This is what you'll be working with. Look at the chart below. If the number is...

1: 1 Always one!
2: 2, 4, 8, 6 Repeating 4 digit even sequence.
3: 3, 9, 7, 1 Repeating 4 digit odd sequence.
4: 4, 6 Only two numbers.
5: 5 Always 5!
6: 6 Always 6, surprisingly for me.
7: 7, 9, 3, 1 Repeating 4 digit odd sequence.
8: 8, 4, 2, 6 Repeating 4 digit even sequence.
9: 9, 1 Only two numbers.
10: 0 Always ends in 0.

11: 1
12: 2, 4, 8, 6
13: 3, 9, 7, 1
14: 4, 6
15: 5

Notice something strange? I do! For one, it repeats after 10. That's why this is so about the last number. Also, for every number 1-10, no matter how many digits the cycle was, the digits in the cycle were always even if the original number was even and vice versa.

Now, after you got your numbers in the sequence, you know what the last numbers will be! Whatever your number is, it is the first number in the sequence. The next number in the sequence occurs when you multiply your number by itself, or to the exponent of two. The last number should be the next number in the sequence. After that, multiply it by itself again. The last number now should be the 3rd number in the sequence. After your sequence is done, it restarts.

You might need an example. Here is one:

1. You pick 7. The sequence is 7, 9, 3, 1!
2. 7x7=49.The 9 is the last number, and is the number in the sequence!
3. Multiply by 7 again. You end up with 343. The 3 fits!
4. 343x7=2,401, which fits the 1!
5. After the sequence is done, it repeats. Next is 16,807, which is the first number in the sequence!

Adding, Subtracting, Multiplying and Dividing Exponents

1. Same base, different exponents add the exponents together. Example is: 8^3 x 8^5=8^8.

2. Have base and exponent in parenthesid and then exponent, multiply exponents together. Example is: (2^3)^5=2^15.

3. Base A with exponent times base B with exponent divided by Base A with different exponent times base B with different exponent equalssubtract exponents from each other. Example is: 3^4 x 4^6 /3^7 x 4^5 = 4^1/3^3=4/3^3=4/27.

4. If base divided by different base with same exponents, then ????. Example is 2^3/3^3.

Area of: Square, Triangle, Circle and Sphere are Related

If base=height on the triangle and that equaled the length in the square, 2T=Sq.
If radius of circle and sphere are the same, 4C=S.
If base and height of triangle are same, if base=5 and the radius in circle=2, areas are similar.
The pattern continues. For every increase in 5 in triangle height, if increase 2 radius, areas remain similar.
In case you didn't figure it out yet, that equals 2.5 to 1. The 1 radius is 3.14. They actually just rounded it modestly. It's actually 3.14159265... Or in other words, π. The triangle with both a base and height of 2.5 have an area of 3.13, just one hundredth from pi!


*KEY*
Sq=Square
T=Triangle
C=Circle
S=Sphere

Baking Soda and Vinegar with Green Food Coloring

I did a science experiment where I mixed baking soda with vinegar. You are probably familiar with this basic experiment of combining acid (vinegar) with alkaline (baking soda). However, I took it even further. I put in the baking soda in my experiment container (a basic jar) and then poured in vinegar. The I added baking soda, then vinegar again... etc. Every time I added the other ingredient the chemical reaction got noticably smaller.

I did another experiment whith the same unwashed glass... only I rinsed out the stuff inside. I put in the baking soda, and dropped in about 6 drops of green food coloring. I tried to mix it in with a spoon, but it wouldn't easily. I poured in the vinegar, and it fizzled and the green coloring concentrated and swirled around in interesting areas. I left it open for about a month. When I came back, the liquid had evaporated, there was the baking soda on the bottom, and on top of the baking soda, there was a broken green crust that looked like the land after a drought, but the cracks were so severe they were completely different pieces. When I poured vinegar onto the baking soda, there was no reaction. I have only one theory of how this could be: When the vinegar evaporated, it left the acid, which cancelled out the alkaline from the baking soda. So, in a sense, I had created baking soda with a neutral Ph! (like water)

Number Palindrome

I found an interesting number palindrome. It works with any number! It's easier with an odd number beggining, as there aren't any decimals, but anything works, even negative numbers! It works like this: Take any number and multiply it by 0. Then take (the number  minus 1) x 0.5 Then (the number minus 2) x 1, and so one, reducing a number on the left by 1 while increasing the one on the right by ½. ***Disclaimer: in negative numbers, start with  the lowest number, then go up by 1 each time, not down. You still raise the right side by ½ however.*** So the rule is really: start with the number farthest from 0 then add/subtract by 1 to get that number closer to 0 every time. The right side of the equation: rules remain the same.

       Positive numbers:

15x0=0             21x0=0         1     Even numbers:

14x0.5=7          20x0.5=10    1

13x1=13           19x1=19       1    4x0=0

12x1.5=18        18x1.5=27    1     3x0.5=1.5

11x2=22           17x2=34       1     2x1=2

10x2.5=25        16x2.5=40    1     1x1.5=1.5

9x3=27             15x3=45       1     0x2=0

8x3.5=28          14x3.5=49    1

7x4=28             13x4=52       +-------------------------------------------------------------------------

6x4.5=27          12x4.5=54    1     Negative numbers:

5x5=25             11x5=55       1

4x5.5=22          10x5.5=55    1     -5x0=0

3x6=18             9x6=54         1     -4x0.5=-2

2x6.5=13          8x6.5=52      1     -3x1=-3

1x7=7               7x7=49         1     -2x1.5=-3

0x7.5=0            6x7.5=45      1     -1x2=-2

                         5x8=40         1      0x2.5=0

                         4x8.5=34      1

                         3x9=27         1

                         2x9.5=19      1

                         1x10=10       1

                         0x10.5=0      1