If you take a lot of numbers and then square them and them subtract the result of each number from the one above it you will notice a pattern.
1²=1 4-1=3
2²=4 9-4=5
3²=9 16-9=7
4²=16 25-16=9
5²=25 36-25=11
6²=36 49-36=13
7²=49 64-49=15
8²=64 81-64=17
9²=81 100-81=19
10²=100
Every time the number you are going up by is increasing by 2. That phrase sounds a lot like my last post. So, let's see if there's something similar for the cubes.
1³=1 8-1=7
2³=8 27-8=19
3³=27 64-27=37
4³=64 125-64=61
5³=125 216-125=91
6³=216 343-216=127
7³=343 512-343=169
8³=512 729-512=217
9³=729 1,000-729=271
10³=1,000
This looks a lot like the exponents from before, except different numbers. So maybe we need to do it again?
19-7=12 127-91=36
37-19=18 169-127=42
61-37=24 217-169=48
91-61=30 271-217=54
It worked! We have the numbers which are increasing by a number again. The difference is that this time it's 6. So it's probably either going up by 4 or multiplying by 3. I'll do one more to see, though I'll probably have to subtract 3 times this time.
14=1 16-1=15 65-15=40 110-40=70 84-70=14
24=16 81-16=65 175-65=110 194-110=84 108-84=20
34=81 256-81=175 369-175=194 302-194=108 132-108=24
44=256 625-256=369 671-369=302 434-302=132 156-132=24
54=625 1,296-625=671 1,105-671=434 590-434=156 180-156=24
64=1,296 2,401-1,296=1,105 1,695-1,105=590 770-590=180 204-180=24
74=2,401 4,096-2,401=1,695 2,465-1,695=770 974-770=204
84=4,096 6,561-4,096=2,465 3,439-2,465=974
94=6,561 10,000-6,561=3,439
104=10,000
Well, I had to do 4 subtractions instead of the 3 I anticipated and something unexpected happened. After 2 numbers it got stuck in a rut of 24. I'll see what happens if I add the 11th number.
114=14,641. 14,641-10,000=4,641. 4,641-3,439=1,202. 1,202-974=228. 228-204=24!
Well, I think it's broken. If you want to do the next stage, do it, but since it might be 8 subtractions, it will be a lot of work. I'm not going to do it, but if you do, please post your findings in a comment for this post.
Friday, September 22, 2017
Wednesday, July 5, 2017
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?
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Hint: The first way deals with exponents and is a geometric sequence.
The other way is by adding and involves an arithmetic sequence.
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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
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
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