Question 1182985: What are the next three terms in the sequence 1,6,2,5,3.....
Found 3 solutions by josgarithmetic, ikleyn, greenestamps:
Answer by josgarithmetic(39613)   (Show Source):
Answer by ikleyn(52754)  (Show Source):
You can put this solution on YOUR website! .
I see another pattern in this post, so my answer is different from that of @josgarithmetic.
I am sure that 5 different people will give 5 different answers, and everybody will be right.
It is because the posed question makes no sense.
The correct answer to the question is "Any number can be next . . . ".
All questions of this type are not mathematical and all are nonsensical.
Please do not post such puzzles to this forum.
They are not Math questions/problems.
Have a nice day (!)
Answer by greenestamps(13195)  (Show Source):
You can put this solution on YOUR website!
It is certainly wrong to say "the next number is 4"... especially since the problem asked for the next THREE numbers.
The easily observable pattern probably seen by the first tutor is of two spliced sequences; one is 1, 2, 3, ... and the other is 6, 5, .... Splicing 1, 2, 3, ... with 6, 5, ... makes the first eight terms of the sequence 1,6,2,5,3,4,4,3,....
But there is nothing in the statement of the problem that tells us that should be the right answer. Other less obvious patterns in the given numbers might lead to very different answers.
And, as the other tutor says, the problem is a puzzle rather than math, because ANY next number will make a valid sequence.
There is, however, a technique that will find "A" mathematically valid answer to any problem like this. But if there is an easily observable pattern, that technique will rarely find the next number produced by that observable pattern.
Any sequence of 5 numbers can be produced by a unique polynomial of degree 4, and by an infinite number of polynomials of higher degrees.
To find the next number in the sequence if the sequence is produced by a polynomial of degree 4, we can use the method of finite differences.
Write the given sequence of numbers....
1 6 2 5 3
Find the differences between successive terms
1 6 2 5 3
5 -4 3 -2
Continue writing rows of differences until the row contains a single number
1 6 2 5 3
5 -4 3 -2
-9 7 -5
16 -12
-28
The single number in the 4th row of differences (the "4th differences" of the original sequence) means the sequence can be generated by a polynomial of degree 4.
To find the next three numbers in the sequence generated by that polynomial, we add 3 more entries in the row of 4th differences showing that the 4th differences are constant.
1 6 2 5 3
5 -4 3 -2
-9 7 -5
16 -12
-28 -28 -28 -28
Then work back up the array to find the next three terms in the original sequence.
1 6 2 5 3 -44 -204 -573
5 -4 3 -2 -47 -160 -369
-9 7 -5 -45 -113 -209
16 -12 -40 -68 -96
-28 -28 -28 -28
One mathematically valid answer: the next three terms in the sequence are -44, -204, and -573.
Quite a bit different than 4, 4, and 3....
|
|
|
