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Hello again, please see my reply to POTW #5 for last week's answers. As always, try to see why or how others' may have obtained different answers than you.
As it goes laterally, and multiplies by 2/term, just calculate (multiplier) to the power of (term) to get the number of numbers on the term number you selected. Divide by 2 to get previous term.
The rightmost number is also the final number/highest in the term in the left to right order, so there would be no subtracting involved.
As the leftmost on the term row is not starting from 0, we must calculate all terms and combine this term, or just calculate the total of the next term and subtract 2, as the term totals before (12th term) make almost the 12th term, excluding the 2 in which the the multiplication started from, so 2 to the power of 12+1, then subtract 2 would be the total amount of numbers, and as the growth of the numbers left to right is +1/number, or x1, so that total would be the same as the rightmost number (biggest) Now the math: 2^12=4096 2048+1024+512+256+128+64+32+16+8+4+2 = *4094 *2 squared starts from 2, yet all numbers here are based on 0, so subtract difference from 2 and 0 (2-4096) 2^13= 8192 *(-2 as starts from 0) 2^13= *8190 The rightmost number is 8190.
What I did to find the answer is quite simple actually. I did an input, output kind of thing. The question asked me to find the right most number in row 12. The fact the question said rightmost, makes this question very simple. Since I don't want to do the whole thing by hand, I had to find the pattern to get from 0 to the next right most number of a row. So I would always choose the number that is on the right, starting with 0. From row 1 to 2, it was 0 then 2, which is add 2. From row 2 to 3, 2 then 6. To get to 6, you need to add 4, double the amount of 2. Before I drew any conclusions, I checked if this would make sense for row 3 to 4. So it would be 6 to 14. To get to 14, I had to add 8, double 4. Now I have pretty good proof that the pattern is start with 0, add 2 and double the amount you add each row. So now the math: Row 1: 0+2=2 Row 2: 2+4=6 Row 3: 6+8=14 Row 4: 14+16=30 5: 30+32=62 6: 62+64=126 7: 126+128= 254 8: 254+256=510 9: 510+512=1022 10:1022+1024=2046 11:2046+2048=4094 12:4094+4096=8190 This means that the right most number on the 12th row is 8190
There are several ways to solve the question. The first one would be to find a pattern in the rightmost numbers and just continue or extend it. I found that in the pattern 0, 2, 6, 14, 30, you could just multiply by 2 and then add 2. I tested this out and found it worked. I then decided to extend the problem using this method for 12 rows. Here is the math: Row 1) 0 2) 0x2+2=2 3) 2x2+2=6 4)6x2+2=14 5)14x2+2=30 6) 30 x2+2=62 7) 62 x 2=124+2=126 8) 126 x 2= 252+2= 254 9) 254x2+2=510 10) 510 x 2+2= 1022 11) 1022x2+2=2046 12) 2046 x 2+2= 4094 If we label: row= r I also can find the value of the 12 row using powers of numbers. I came up with the formula 2˄r-2= value or rightmost number for row. value = 2˄r - 2 when r = 1 value = 0 as 2x1-2=0 r = 2 value = 2 as 2x2-2=2... r = 3 value = 6 r=4 value=14 I can fast forward to finding the twelfth row and do 2˄12-2= 4094.
row # # of integers 1 1 2 2 3 4 4 8 5 16 6 32 7 64 8 128 9 256 10 512 11 1024 12 2048 if i add all the numbers i will get my answer i have to subtract one because row number one is not 1 4095 then minus one is 4094
To figure out the answer, I just found a pattern in the rightmost section of the web. It was much easier and efficient than copying and completing the whole web. I found more than one pattern, but decided to use "Multiply by 2 and then add 2 every time", as that is an easy rule to follow. I did this 8 times, as it was asking for the 12th row and 4 rows were completed. In the end, I got 8190 in the 12th row.
I found out the pattern for the right part of the graph. At first I was thinking that I would have to draw the whole diagram, but then I thought about it and came to this much simpler and faster solution. I found the pattern times 2 then add 2, tested it, and found that it worked. I did all the math and got the answer of 4094 at the right of the 12th row.
I just need the find the pattern for the chart 0 1 2 x2 which is 4 so 3 4 5 6 and then keep on multiplying until you get the 12th row or do what I did which is to multiply the last number by 2 and then add to to get 8190 at the 12th row for the most right number
First I looked at the picture and tried to find a pattern, after a while, I realized that rightmost “column” was defined with a *2+2 pattern rule. I checked if I was correct: 0*2+2=2 2*2+2+6 6*2+2=14. I then put the rule to use. The “-” symbolizes the numbers left of it.
The first thing I did was write down the next 2 rows (5th and 6th), but that's when I noticed a pattern. The pattern was that every row the right most number was just multiplied by 2 and then add 2. 0 2 6 14 30 62 126 254 510 1022 2046 4094 The rightmost number on the 12th row is 4094
As with all tricky problems that are grade appropriate, there is always a pattern. So I looked for one. The rightmost number on the 12th row is what the question asked for. I started at the diagram for a while and noticed some key things.
1. The number of numbers in each row doubles each time meaning that the 12th row is 2 doubled 12 times.
2. 2(exponent the row number) + the outcome of the term before it= the last number of the row
3. the last number of the row is 2(exponent the row number)-2
This means that this equation solves the problem: 2 (2 to the power of 12)- 2= the answer.
2 x (2 to the power of 12) = 2 x 4096 2 x4096= 8192 8192-2=8190
and the final answer is: The right-most number of the 12th row is 8190.
At first I tried to draw the graph but after I realized that would take up too much space I tried looking for the pattern. The pattern on the far right of the graph is x2+2. 0 0x2+2=2 2x2+2=6 6x2+2=14 14x2+2=30 30x2+2=62 62x2+2=126 126x2+2=254 254x2+2=510 510x2+2=1022 1022x2+2=2046 2046x2+2=4094 Finally I got my answer of 4094
I tried to find a pattern with these numbers. And like most patterns I decided it would involve multiplication, division, addition and substration. So I tried to find one. I found *2+2. Then I decided to test it out. So I found 0*2+2=2, 2*2+2=6, 6*2+2=14, 14*2+2=30. Then I was pretty sure the pattern worked. So I continued. number 1=0 number 2=2 number 3=6 number 4=14 number 5=30 number 6=62 number 7=126 number 8=254 number 9=510 number 10=1022 number 11=2046 number 12=4094
The pattern from 0 to 14 is times 2 +2 because: (0x2)+2=2, (2x2)+2=6, (6x2)+2=14 There are 12 rows and 14 is the rightmost in the 4th row so we have 8 more rows to go:
Row 1: 0 Row 2: (0 x 2) + 2 =2 Row 3: (2 x 2) + 2 =6 Row 4: (6 x 2) + 2 =14 Row 5: (14 x 2) + 2 =30 Row 6: (30 x 2) + 2 =62 Row 7: (62 x 2) + 2 =126 Row 8: (126 x 2) + 2 =254 Row 9: (254 x 2) + 2 =510 Row 10:(510 x 2) + 2 =1022 Row 11: (1022 x 2) + 2 =2046 Row 12:(2046 x 2) + 2 =4094 The rightmost number in row 12 is 4094.
To solve this POTW, I first saw that I only had to find the right-most number on the 12th row. So, I only had to look at the right-most number in each row to find a pattern. The pattern is that the difference between each rightmost number in each row is the previous sum plus the previous number that was added (the second one)* 2. So, I continued this until I had 12 numbers.
The answer I got was that the most right number in the 12th row is 4094.
How I solved the problem was that I first looked for the pattern that was being used on the very right of the diagram. I noticed that the pattern increased by having the number be multiplied by 2 and the have 2 added to the number or x 2 + 2. To get the answer you keep on going down in the problem by using that problem each time.
Row 1: 0 Row 2: 0 x 2 + 2 = 2 Row 3: 2 x 2 + 2 = 6 Row 4: 6 x 2 + 2 = 14 Row 5: 14 x 2 + 2 = 30 Row 6: 30 x 2 + 2 = 62 Row 7: 62 x 2 + 2 = 126 Row 8: 126 x 2 + 2 = 254 Row 9: 254 x 2 + 2 = 510 Row 10: 510 x 2 + 2 = 1022 Row 11: 1022 x 2 + 2 = 2046 Row 12: 2046 x 2 + 2 = 4094
Therefore, the most right number in the 12th row is 4094.
One approach to solving the problem would be to write out the first 12 rows of the chart and read off the rightmost number in row 12. You would discover that the number is 4094. This solution may “work” in this example but it is certainly not ideal. It would not be practical if you were asked for the last number in row 50.
Another way: This solution only looks at the rightmost number in each row. To get from the top number to the rightmost number in row 2 add 2. To get from the rightmost number in row 2 to the rightmost number in row 3 add 4. To get from the rightmost number in row 3 to the rightmost number in row 4 add 8. These numbers which are added correspond to the number of numbers in the next row. We must add 11 of these numbers to 0. 0 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + 512 + 1024 + 2048 = 4094 The rightmost number in row 12 is 4094.
To solve this problem, I first did a few more rows. I quickly discovered the pattern in the rightmost number was times 2 plus 2 each time, so I continued this until the 12th row. 0 2 6 14 30 62 126 254 510 1022 2046 4094 Therefore, the rightmost number in row 12 is 4094.
I did it on paper
ReplyDeleteAs it goes laterally, and multiplies by 2/term, just calculate (multiplier) to the power of (term) to get the number of numbers on the term number you selected. Divide by 2 to get previous term.
ReplyDeleteThe rightmost number is also the final number/highest in the term in the left to right order, so there would be no subtracting involved.
As the leftmost on the term row is not starting from 0, we must calculate all terms and combine this term, or just calculate the total of the next term and subtract 2, as the term totals before (12th term) make almost the 12th term, excluding the 2 in which the the multiplication started from, so 2 to the power of 12+1, then subtract 2 would be the total amount of numbers, and as the growth of the numbers left to right is +1/number, or x1, so that total would be the same as the rightmost number (biggest)
Now the math:
2^12=4096
2048+1024+512+256+128+64+32+16+8+4+2 = *4094
*2 squared starts from 2, yet all numbers here are based on 0, so subtract difference from 2 and 0 (2-4096)
2^13= 8192 *(-2 as starts from 0)
2^13= *8190
The rightmost number is 8190.
What I did to find the answer is quite simple actually. I did an input, output kind of thing. The question asked me to find the right most number in row 12. The fact the question said rightmost, makes this question very simple. Since I don't want to do the whole thing by hand, I had to find the pattern to get from 0 to the next right most number of a row. So I would always choose the number that is on the right, starting with 0. From row 1 to 2, it was 0 then 2, which is add 2. From row 2 to 3, 2 then 6. To get to 6, you need to add 4, double the amount of 2. Before I drew any conclusions, I checked if this would make sense for row 3 to 4. So it would be 6 to 14. To get to 14, I had to add 8, double 4. Now I have pretty good proof that the pattern is start with 0, add 2 and double the amount you add each row. So now the math:
ReplyDeleteRow 1: 0+2=2
Row 2: 2+4=6
Row 3: 6+8=14
Row 4: 14+16=30
5: 30+32=62
6: 62+64=126
7: 126+128= 254
8: 254+256=510
9: 510+512=1022
10:1022+1024=2046
11:2046+2048=4094
12:4094+4096=8190
This means that the right most number on the 12th row is 8190
There are several ways to solve the question. The first one would be to find a pattern in the rightmost numbers and just continue or extend it. I found that in the pattern 0, 2, 6, 14, 30, you could just multiply by 2 and then add 2. I tested this out and found it worked. I then decided to extend the problem using this method for 12 rows. Here is the math:
ReplyDeleteRow 1) 0
2) 0x2+2=2
3) 2x2+2=6
4)6x2+2=14
5)14x2+2=30
6) 30 x2+2=62
7) 62 x 2=124+2=126
8) 126 x 2= 252+2= 254
9) 254x2+2=510
10) 510 x 2+2= 1022
11) 1022x2+2=2046
12) 2046 x 2+2= 4094
If we label:
row= r
I also can find the value of the 12 row using powers of numbers. I came up with the formula 2˄r-2= value or rightmost number for row.
value = 2˄r - 2
when
r = 1 value = 0 as 2x1-2=0
r = 2 value = 2 as 2x2-2=2...
r = 3 value = 6
r=4 value=14
I can fast forward to finding the twelfth row and do 2˄12-2= 4094.
To find out what the number to the very right in the 12th row, I tried to find a pattern between the last number of each row
ReplyDelete0 (Row 1)
0 + 2 = 2 (Row 2)
2 + 4 (2X2) = 6 (Row 3)
6 + 8 (4X2) = 14 (Row 4)
14 + 16 (8X2) = 30 (Row 5)
30 + 32 (16X2) = 62 (Row 6)
62 + 64 (32X2) = 126 (Row 7)
126 + 128 (64X2) = 254 (Row 8)
254 + 256 (128X2) = 510 (Row 9)
510 + 512 (256X2) = 1022 (Row 10)
1022 + 1024 (512X2) = 2046 (Row 11)
2046 + 2048 (1024X2) = 4094 (Row 12)
The last number in the 12th row (left to right) is 2094
Sorry, typo, I meant 4094 for my answer
ReplyDeleteanswer
ReplyDeleterow #
# of integers
1 1
2 2
3 4
4 8
5 16
6 32
7 64
8 128
9 256
10 512
11 1024
12 2048
if i add all the numbers i will get my answer
i have to subtract one because row number one is not 1
4095 then minus one is
4094
the game 2048 helped
To figure out the answer, I just found a pattern in the rightmost section of the web. It was much easier and efficient than copying and completing the whole web. I found more than one pattern, but decided to use "Multiply by 2 and then add 2 every time", as that is an easy rule to follow. I did this 8 times, as it was asking for the 12th row and 4 rows were completed. In the end, I got 8190 in the 12th row.
ReplyDeleteI think I accidentally added an extra row when I did my calculations
DeleteI found out the pattern for the right part of the graph. At first I was thinking that I would have to draw the whole diagram, but then I thought about it and came to this much simpler and faster solution. I found the pattern times 2 then add 2, tested it, and found that it worked. I did all the math and got the answer of 4094 at the right of the 12th row.
ReplyDeleteI just need the find the pattern for the chart
ReplyDelete0
1 2 x2 which is 4 so 3 4 5 6 and then keep on multiplying until you get the 12th row or do what I did which is to multiply the last number by 2 and then add to to get 8190 at the 12th row for the most right number
First I looked at the picture and tried to find a pattern, after a while, I realized that rightmost “column” was defined with a *2+2 pattern rule. I checked if I was correct: 0*2+2=2 2*2+2+6 6*2+2=14. I then put the rule to use. The “-” symbolizes the numbers left of it.
ReplyDelete0
- 2
- 6
- 14
- 30
- 62
- 126
- 254
- 510
- 1022
- 2046
- 4094
Therefore, the rightmost number in row 12 is 4094.
The first thing I did was write down the next 2 rows (5th and 6th), but that's when I noticed a pattern. The pattern was that every row the right most number was just multiplied by 2 and then add 2.
ReplyDelete0
2
6
14
30
62
126
254
510
1022
2046
4094
The rightmost number on the 12th row is 4094
As with all tricky problems that are grade appropriate, there is always a pattern.
ReplyDeleteSo I looked for one.
The rightmost number on the 12th row is what the question asked for.
I started at the diagram for a while and noticed some key things.
1. The number of numbers in each row doubles each time meaning that the 12th row is 2 doubled 12 times.
2. 2(exponent the row number) + the outcome of the term before it= the last number of the row
3. the last number of the row is 2(exponent the row number)-2
This means that this equation solves the problem:
2 (2 to the power of 12)- 2= the answer.
2 x (2 to the power of 12) = 2 x 4096
2 x4096= 8192
8192-2=8190
and the final answer is: The right-most number of the 12th row is 8190.
I really hope that made sense.
At first I tried to draw the graph but after I realized that would take up too much space I tried looking for the pattern. The pattern on the far right of the graph is x2+2.
ReplyDelete0
0x2+2=2
2x2+2=6
6x2+2=14
14x2+2=30
30x2+2=62
62x2+2=126
126x2+2=254
254x2+2=510
510x2+2=1022
1022x2+2=2046
2046x2+2=4094
Finally I got my answer of 4094
I tried to find a pattern with these numbers. And like most patterns I decided it would involve multiplication, division, addition and substration. So I tried to find one. I found *2+2. Then I decided to test it out. So I found 0*2+2=2, 2*2+2=6, 6*2+2=14, 14*2+2=30. Then I was pretty sure the pattern worked. So I continued.
ReplyDeletenumber 1=0
number 2=2
number 3=6
number 4=14
number 5=30
number 6=62
number 7=126
number 8=254
number 9=510
number 10=1022
number 11=2046
number 12=4094
Here’s how I solved the problem:
ReplyDeleteThe pattern from 0 to 14 is times 2 +2 because:
(0x2)+2=2, (2x2)+2=6, (6x2)+2=14
There are 12 rows and 14 is the rightmost in the 4th row so we have 8 more rows to go:
Row 1: 0
Row 2: (0 x 2) + 2 =2
Row 3: (2 x 2) + 2 =6
Row 4: (6 x 2) + 2 =14
Row 5: (14 x 2) + 2 =30
Row 6: (30 x 2) + 2 =62
Row 7: (62 x 2) + 2 =126
Row 8: (126 x 2) + 2 =254
Row 9: (254 x 2) + 2 =510
Row 10:(510 x 2) + 2 =1022
Row 11: (1022 x 2) + 2 =2046
Row 12:(2046 x 2) + 2 =4094
The rightmost number in row 12 is 4094.
To find the end number in the 12th row, I tried finding every pattern until I got to 12th.
ReplyDeleteRow 1: 0
Row 2: 0 + 2 = 2
Row 3: (2x2) 2 + 4 = 6
Row 4: (4x2) 6 + 8 = 14
Row 5: (8x2)14 + 16 = 30
Row 6: (16x2)30 + 32 = 62
Row 7: (32x2)62 + 64 = 126
Row 8: (64x2) 126 + 128 = 254
Row 9: (128x2) 254 + 256 = 510
Row 10: (256x2) 510 + 512 = 1022
Row 11: (512x2) 1022 + 1024 = 2046
Row 12: (1024x2) 2046 + 2048 = 4094 < Is the the most right number in the 12th row
To solve this POTW, I first saw that I only had to find the right-most number on the 12th row. So, I only had to look at the right-most number in each row to find a pattern. The pattern is that the difference between each rightmost number in each row is the previous sum plus the previous number that was added (the second one)* 2. So, I continued this until I had 12 numbers.
ReplyDeleteRow 1: 0
Row 2: 2
+2+ 4= 6
Row 3: 6
+6+ 8= 14
Row 4: 14
+14+16= 30
Row 5: 30
+30+32=62
Row 6: 62
62+64= 126
Row 7: 126
126+128= 254
Row 8: 254
254+256= 510
Row 9: 510
510+512= 1022
Row 10: 1022
1022+1024=2046
Row 11:2046
2046+2048=4094
Row 12: 4096
Therefore, the rightmost number in row 12 would be 4096.
The answer I got was that the most right number in the 12th row is 4094.
ReplyDeleteHow I solved the problem was that I first looked for the pattern that was being used on the very right of the diagram. I noticed that the pattern increased by having the number be multiplied by 2 and the have 2 added to the number or x 2 + 2. To get the answer you keep on going down in the problem by using that problem each time.
Row 1: 0
Row 2: 0 x 2 + 2 = 2
Row 3: 2 x 2 + 2 = 6
Row 4: 6 x 2 + 2 = 14
Row 5: 14 x 2 + 2 = 30
Row 6: 30 x 2 + 2 = 62
Row 7: 62 x 2 + 2 = 126
Row 8: 126 x 2 + 2 = 254
Row 9: 254 x 2 + 2 = 510
Row 10: 510 x 2 + 2 = 1022
Row 11: 1022 x 2 + 2 = 2046
Row 12: 2046 x 2 + 2 = 4094
Therefore, the most right number in the 12th row is 4094.
x
ReplyDeleteOne approach to solving the problem would be to write out the first 12 rows of the chart and
ReplyDeleteread off the rightmost number in row 12. You would discover that the number is 4094. This
solution may “work” in this example but it is certainly not ideal. It would not be practical if
you were asked for the last number in row 50.
Another way: This solution only looks at the rightmost number in each row.
To get from the top number to the rightmost number in row 2 add 2. To get from the
rightmost number in row 2 to the rightmost number in row 3 add 4. To get from the rightmost
number in row 3 to the rightmost number in row 4 add 8. These numbers which are added
correspond to the number of numbers in the next row. We must add 11 of these numbers to 0.
0 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + 512 + 1024 + 2048 = 4094
The rightmost number in row 12 is 4094.
To solve this problem, I first did a few more rows. I quickly discovered the pattern in the rightmost number was times 2 plus 2 each time, so I continued this until the 12th row.
ReplyDelete0
2
6
14
30
62
126
254
510
1022
2046
4094
Therefore, the rightmost number in row 12 is 4094.