SOLUTION: 51, 53, 56, 60 what comes next? 51, 53, 56, 60, __, __

Algebra ->  Sequences-and-series -> SOLUTION: 51, 53, 56, 60 what comes next? 51, 53, 56, 60, __, __      Log On


   

Question 1186602: 51, 53, 56, 60 what comes next? 51, 53, 56, 60, __, __
Found 4 solutions by MathLover1, Alan3354, ikleyn, greenestamps:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
sequence : 51.....53....56.....60
First diff. : .....2......3.....4
Second diff. : ......1....1
We see that the second differences (blue ones) are all equal so we concludet that this is a quadratic sequence.
The quadratic sequence has the form T%5Bn%5D=an%5E2%2Bbn%2Bc
To find the value of a we just divide second difference ( 1 ) by 2.

a=1%2F2
Now we have: T%5Bn%5D=%281%2F2%29n%5E2%2Bbn%2Bc
Substitute n=1 and n=2 into above equation:
T%5B1%5D=%281%2F2%291%5E2%2Bb%2A1%2Bc
T%5B2%5D=%281%2F2%292%5E2%2Bb%2A2%2Bc
---------------------------------------- Since T%5B1%5D=51 and T%5B2%5D=53, we have
51=1%2F2%2Bb%2Bc....eq.1
53=2%2B2b%2Bc.............e.2
---------------------------------------subtract eq.1 from eq.2

53-51=2%2B2b%2Bc-%281%2F2%2Bb%2Bc%29
2=2%2B2b%2Bc-1%2F2-b-c
2=3%2F2%2Bb....... solve for b
2-3%2F2=b
b=1%2F2
go to
51=1%2F2%2Bb%2Bc....eq.1, substitute+b
51=1%2F2%2B1%2F2%2Bc....... solve for c
51=1%2Bc
c=51-1
c=50
and, your nth term formula is: T%5Bn%5D=%281%2F2%29n%5E2%2B%281%2F2%29n%2B50
then next term will be 5th term, means n=5
T%5B5%5D=%281%2F2%295%5E2%2B%281%2F2%295%2B50
T%5B5%5D=65

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
you can pick any numbers and they can be justified.
problems like this are a waste of time.

Answer by ikleyn(52790) About Me  (Show Source):
You can put this solution on YOUR website!
.

I can construct a polynomial of high degree, which will has given values in given points,
and totally different value in the next point.


An arbitrary person from the street can construct a polynomial even more hire degree
and obtain different  "next"  number.


As the question is posed in the post,  IT  MAKES  the problem non-Mathematical.


With such questions, you should go to fortune-tellers.



Actually,  such questions is the way for them to make money.


More reasonable/educated people do not play such games . . . And do not ask such questions . . .


////////////


"What is the pattern ?"   would be correct,  reasonable question in this context.

Even better is to ask  "what pattern do you see ?",  because another person can see different pattern.

"What is the next number",  or   "what comes next"   is not correct or reasonable question in this context.


            ANY  NUMBER  CAN  BE  NEXT,  if restrictions are not imposed.


Also,  to ask  "predict next number,  using a polynomial of lowest degree"  is a correct question
in this context,  because it determines a procedure by an unique way.



        . . . Actually,  at the level of common sense,  all these things are  SELF - EVIDENT . . .

                        . . . They do not require special knowledge . . .



Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Contrary to what two other tutors have said, it is not always true that this kind of problem is a waste of time.

It is indeed true that any next number will make a valid sequence, as one tutor said.

It is also true, as another tutor said, that we can find as many polynomial functions as we want that produce the given first four numbers and produce different subsequent numbers. So even with formal mathematical methods we can get an infinite number of different results for the next two numbers in the sequence.

However, those facts do not make all problems like this a waste of time.

Mathematics is full of wonderful (and wondrous) patterns. Good problems like this help teach a student to look for patterns.

At this forum (and elsewhere) we see a great many problems like this that have no easily discernible pattern; THOSE problems are usually a waste of time.

But this example has an obvious pattern which is PROBABLY the expected answer. True, we can't KNOW that it is the expected answer; but we can guess that it probably is.

The pattern is that the differences between successive terms of the sequence increase by 1:
51 + 2 = 53
53 + 3 = 56
56 + 4 = 60

Continue that pattern to find as many more terms of the sequence as you want:
60 + 5 = 65
65 + 6 = 71
71 + 7 = 78
etc...