course Mth 151 I mislabeled this earlier as 11 text, sorry. ?????????O?assignment #006006. Sequences and Patterns
......!!!!!!!!...................................
19:11:12 `q001. Note that there are 6 questions in this assignment. Find the likely next element of the sequence 1, 2, 4, 7, 11, ... .
......!!!!!!!!...................................
RESPONSE --> Using inductive reasoning, one can assume that the next likely element is 16, as the pattern it shows us increases its additive properties by 1 with each increasing number. So 16 is the most likely unless it's in a different context from counting numbers. confidence assessment: 3
.................................................
......!!!!!!!!...................................
19:11:19 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.
......!!!!!!!!...................................
RESPONSE --> OK self critique assessment: 3
.................................................
......!!!!!!!!...................................
19:16:09 `q002. Find the likely next two elements of the sequence 1, 2, 4, 8, 15, 26, ... .
......!!!!!!!!...................................
RESPONSE --> 42 and 64. Using inductive reasoning and finding the pattern, 1+1=2 2+2=4 4+4=8 8+7=15 and so on. The pattern is steadily increasing, as you add the next counting number to the sum of what you get when you add. This could be completely wrong, but this is how I arrived at it. confidence assessment: 2
.................................................
......!!!!!!!!...................................
19:17:47 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, ... .
......!!!!!!!!...................................
RESPONSE --> I arrived at the answer in an entirely incorrect fashion. At first I was trying to find differences, but for some reason never arrived at a correct conclusion, in my head or in paper. Once again, it's a dilema of simply not going deep enough into the problem, and looking at all angles. self critique assessment: 2
.................................................
......!!!!!!!!...................................
19:19:17 `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.
......!!!!!!!!...................................
RESPONSE --> 16. This is one of the previous possibilities I came up with in attempting to solve the previous problem. This inductive pattern multiplies every element by 2 to arrive at the next element. 1*2=2 2*2=4 4*2=8 8*2=16 confidence assessment: 3
.................................................
......!!!!!!!!...................................
19:19:33 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, ... .
......!!!!!!!!...................................
RESPONSE --> OK self critique assessment: 3
.................................................
......!!!!!!!!...................................
19:21:41 09-16-2008 19:21:41 `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.
......!!!!!!!!...................................
NOTES -------> The sequence of ratios is 1.5. So using this result to estimate the next number and sequence gives us 162, 243, 364.5.....
.......................................................!!!!!!!!...................................
19:21:43 `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.
......!!!!!!!!...................................
RESPONSE --> confidence assessment: 3
.................................................
......!!!!!!!!...................................
19:21:52 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.
......!!!!!!!!...................................
RESPONSE --> OK self critique assessment: 3
.................................................
......!!!!!!!!...................................
19:26:21 `q005. Find the sequence of ratios for the sequence 1, 2, 3, 5, 8, 13, 21... , and estimate the next element of the sequence.
......!!!!!!!!...................................
RESPONSE --> The sequence of ratios is very strange, as it doesn't follow a specific pattern. 2/1= 2 3/2=1.5 5/3=1.6 13/8= 1.6 21/13= 1.6 Using this, one can only assume that since the past couple of numbers were around the ballpark of 1.6, 33 or 34 will be the next element. confidence assessment: 2
.................................................
......!!!!!!!!...................................
19:27:29 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, ... .
......!!!!!!!!...................................
RESPONSE --> OK, I will remember to do better rounding with my decimal answers, but I am happy I arrived at that conclusion by rounding the correct last element. self critique assessment: 2
.................................................
......!!!!!!!!...................................
19:28:39 `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?
......!!!!!!!!...................................
RESPONSE --> 55, 89, 144. It's simply adding the numbers together in pairs, and using that sum to find the next element. confidence assessment: 3
.................................................
......!!!!!!!!...................................
19:28:47 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, ... .
......!!!!!!!!...................................
RESPONSE --> OK self critique assessment: 3
.................................................