course Mth 151 ~wc|NVzwassignment #006
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12:23:10 `q001. Note that there are 6 questions in this assignment. Find the likely next element of the sequence 1, 2, 4, 7, 11, ... .
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RESPONSE --> To find the likely next element of the sequence 1, 2, 4, 7, 11, ... we must first find the pattern. We see that if we take the previous number in the pattern and add it to the next number in our natural counting number (exp..1,2,3,4,5....) we will get the next number in the sequence. 1, 2, 4, 7, 11, ... 1+1=2 2+2=4 4+3=7 7+4=11 11+5=16 so our next element in this sequence would be 16 confidence assessment: 3
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12:23:43 The difference between 1 and 2 is 1; between 2 and 4 is 2; between 4 and 7 is 3; between 7 and 11 is 4. So we expect that the next difference will be 5, which will make the next element 11 + 5 = 16.
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RESPONSE --> I answered this correctly self critique assessment: 3
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12:53:08 `q002. Find the likely next two elements of the sequence 1, 2, 4, 8, 15, 26, ... .
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RESPONSE --> To find the likely next two elements of the sequence 1, 2, 4, 8, 15, 26, ... first we need to find the difference between the numbers we have in our sequence. So the differences are 2-1=1 4-2=2 8-4=4 15-8=7 26-15=11 We can see a pattern in our answers. The difference between each answer goes up by the next natural counting number. so if we add the next natural counting number to last difference we will get the difference between the next number and previous. We will use x to stand for the unknown number and evaluate the problem. 2-1=1+1=2 4-2=2+2=4 8-4=4+3=7 15-8=7+4=11 26-15=11+5=16 x - 26=16= x=16+26 x=42 x - 42=22 x=22+42 x=64 So the next 2 elements in this sequence would likely be 42 & 64 confidence assessment: 2
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13:02:02 The difference between 1 and 2 is 1; the difference between 2 and 4 is 2, the difference between 4 and 8 is 4; the difference between 8 and 15 is 7; the difference between 15 and 26 is 11. The differences form the sequence 1, 2, 4, 7, 11, ... . As seen in the preceding problem the differences of this sequence are 1, 2, 3, 4, ... . We would expect the next two differences of this last sequence to be 5 and 6, which would extend the sequence 1, 2, 4, 7, 11, ... to 1, 2, 4, 7, 11, 16, 22, ... . If this is the continuation of the sequence of differences for the original sequence 1, 2, 4, 8, 15, 26, ... then the next two differences of this sequence would be 16 , giving us 26 + 16 = 42 as the next element, and 22, giving us 42 + 26 = 68 as the next element. So the original sequence would continue as 1, 2, 4, 8, 15, 26, 42, 68, ... .
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RESPONSE --> I am a little confused about the number in the sequence. In my answer I got the next 2 elements would be 42 & 64 and your answers are 42 & 68. Isn't it true that we do not know the next number but we know the difference between the next number and the previous number is 22 so wouldnt that be x - 42=22 x=22+42 x=64 If I am wrong please show me how I messed this up. self critique assessment: 2
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13:09:08 `q003. What would be the likely next element in the sequence 1, 2, 4, 8, ... . It is understood that while this sequence starts off the same as that in the preceding exercise, it is not the same. The next element is not 15, and the pattern of the sequence is different than the pattern of the preceding.
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RESPONSE --> To find the likely next element in the sequence 1, 2, 4, 8 we need to find the difference between the number 2-1=1 4-2=2 8-4=4 we can see that the difference doubles each time. x-8=8 x=8+8 x=16 x-16=16 x=16+16 x=32 so the next likely elements are 16 & 32 confidence assessment: 3
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13:09:39 One obvious pattern for this sequence is that each number is doubled to get the next. If this pattern continues then the sequence would continue by doubling 8 to get 16. The sequence would therefore be 1, 2, 4, 8, 16, ... .
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RESPONSE --> I answered this correctly self critique assessment: 0
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13:22:11 `q004. There are two important types of patterns for sequences, one being the pattern defined by the differences between the numbers of the sequence, the other being the pattern defined by the ratios of the numbers of the sequence. In the preceding sequence 1, 2, 4, 8, 16, ..., the ratios were 2/1 = 2; 4/2 = 2; 8/4 = 2; 16/8 = 2. The sequence of ratios for 1, 2, 4, 8, 16, ..., is thus 2, 2, 2, 2, a constant sequence. Find the sequence of ratios for the sequence 32, 48, 72, 108, ... , and use your result to estimate the next number and sequence.
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RESPONSE --> To find the sequence of ratios for the sequence 32, 48, 72, 108 we must divide the next number from the previous number in the sequence. 48/32=1.5 72/48=1.5 108/72=1.5 Lets evaluate x/108=1.5 x=1.5*108 x=162 So the ratio is 1.5 and the next number in the sequence is 162 confidence assessment: 3
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13:22:28 The ratios are 48/32 = 1.5; 72 / 48 = 1.5; 108/72 = 1.5, so the sequence of ratios is 1.5, 1.5, 1.5, 1.5, ... . The next number the sequence should probably therefore be 108 * 1.5 = 162.
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RESPONSE --> I answered this correctly self critique assessment: 3
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13:33:15 `q005. Find the sequence of ratios for the sequence 1, 2, 3, 5, 8, 13, 21... , and estimate the next element of the sequence.
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RESPONSE --> This question has stumped me I am moving on to the answer confidence assessment: 0
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13:45:40 The ratios are 2/1 = 2; 3/2 = 1.5; 5/3 = 1.66...; 8/5 = 1.60; 13/8 = 1.625; 21/13 = 1.615. The sequence of ratios is 2, 1.5, 1.66..., 1.625, 1.615, ... . We see that each number in the sequence lies between the two numbers that precede it -- 1.66... lies between 2 and 1.5; 1.60 lies between 1.5 and 1.66...; 1.625 lies between 1.66... and 1.60; 1.615 lies between 1.60 and 1.625. We also see that the numbers in the sequence alternate between being greater than the preceding number and less than the preceding number, so that the intervals between the numbers get smaller and smaller. So we expect that the next number in the sequence of ratios will be between 1.615 and 1.625, and if we pay careful attention to the pattern we expect the next number to be closer to 1.615 than to 1.625. We might therefore estimate that the next ratio would be about 1.618. We would therefore get 1.618 * 21 = 33.98 for the next number in the original sequence. However, since the numbers in the sequence are all whole numbers, we round our estimate up to 34. Our conjecture is that the sequence continues with 1, 2, 3, 5, 8, 13, 21, 34, ... .
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RESPONSE --> Yes I see the method now. I was confused because when dividing the next number by the previous number I was not getting the same number or whole numbers. I was on the right track in doing the problem on paper but thought I was wrong for that reason. self critique assessment: 2
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14:13:27 `q006. Without using ratios, can you find a pattern to the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, ..., and continue the sequence for three more numbers?
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RESPONSE --> First to find a pattern to the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, ... we need to see the difference between the previous number and the next number in the pattern 1-1=0 2-1=1 3-2=1 5-3=2 8-5=3 13-8=5 21-13=8 34-21=13 We see that the difference between the next number and the previous number is number before those two numbers so we know that x-34=21 x=21+34 x=55 x-55=34 x=34+55 x=89 x-89=55 x=55+89 x=144 So the next 3 numbers in the sequence is 55,89,144 confidence assessment: 3
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14:14:24 The pattern is that each element from the third on is the sum of the two elements that precede it. That is, 1+1=2, 2+1=3; 3+2=5; 5+3=8; 8+5=13; 13+8=21; 21+13=34; . The next three elements would therefore e 34+21=55; 55+34=89; 89+55=144. . The sequence is seen to be 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... .
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RESPONSE --> I answered this correctly self critique assessment: 3
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