Number series

Here is one site I found that has this question (with additional options): #21 in https://gradeup.co/liveData/f/2019/7/english-part-45.pdf

They are all presented as if there were one correct answer, and one could know "the pattern". This is not mathematics.

I generally try not to belittle the education establishment of other countries, but if this is what they are teaching, they are not preparing students to do anything useful.
 
Jeff is correct when he said There is one (pattern)that will work for every proposed answer.
 
Dr.Peterson said:
Here is one site I found that has this question (with additional options): #21 in https://gradeup.co/liveData/f/2019/7/english-part-45.pdf

They are all presented as if there were one correct answer, and one could know "the pattern". This is not mathematics.

I generally try not to belittle the education establishment of other countries, but if this is what they are teaching, they are not preparing students to do anything useful.
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I believe these are "challenge" problem (for extra credit) - where often the questioner provides a foot-note "If you can solve this problem - you are better than me!"

As to the infinite solutions - in general those solutions will not provide the "exact" integer output.

I am guessing this - because I was "pressed through" that torture mill.
 
The On-Line Encyclopedia of Integer Sequences SEE HERE is the largest collection of integer sequences in existence. It was founded in 1964.
I copied the post '7, 9, 12, 48, ?, 890' pasted it into that site replacing the ? with each of the four options. In each case there was no corresponding sequence on record. That makes me think that this question is not worthwhile to work on. I cannot think of a time when I did not find at least one sequence which matched part of the sequence about which I asked.
 
nasi112 said:
This is not a pattern. A pattern is something that you can express with a sequence formula or summation formula.

Something like this

[MATH]a_k = \frac{(-1)^k}{k3^k}[/MATH]
A pattern means that there is a relationship between [MATH]a_{96}[/MATH] and [MATH]a_{97}[/MATH] with one formula.

Therefore, give me one single formula that generates those numbers you have given, then I will agree with you it has a pattern. Otherwise, it is just a game of imagination.
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Actually yes, the pattern is consistent in the sense that when we are traveling from the odd term to even term everytime we are adding and from even to odd we are subtracting.

But the Syntax is same but addition and subtraction Will alternatively put into effect
 
Dr.Peterson said:
Here is one site I found that has this question (with additional options): #21 in https://gradeup.co/liveData/f/2019/7/english-part-45.pdf

They are all presented as if there were one correct answer, and one could know "the pattern". This is not mathematics.

I generally try not to belittle the education establishment of other countries, but if this is what they are teaching, they are not preparing students to do anything useful.
Click to expand...
Give me a pattern or syntax that gives rise to other proposed Answers except 172.
 
Subhotosh Khan said:
I believe these are "challenge" problem (for extra credit) - where often the questioner provides a foot-note "If you can solve this problem - you are better than me!"

As to the infinite solutions - in general those solutions will not provide the "exact" integer output.

I am guessing this - because I was "pressed through" that torture mill.
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These are puzzles, not math problems. Yes, often one can, with perseverance, find a "pattern" that produces it without resorting to fifth degree polynomials and the like, but that is a "skill" that is worthless. It does not make one person "better" than another, except for being more willing to waste the time.

Yes, the coefficients of the polynomial are often irrational, so that the result is not exact; but not always. One such puzzle you asked of me privately was

Find the value of X in this sequence:​
11,6,5,9,16, X​
Options:​
66.5​
78.5​
89.5​
42.5​
31.5​

I found the formula a_n = -1/8*n^4 + 17/12*n^3 - 27/8*n^2 - 35/12*n + 16. Using this formula, the 6th term turns out to be 21, exactly. This is not one of the choices, but is considerably "nicer" than any of them. And what were you saying about "exact integer output"?

pka said:
The On-Line Encyclopedia of Integer Sequences SEE HERE is the largest collection of integer sequences in existence. It was founded in 1964.
I copied the post '7, 9, 12, 48, ?, 890' pasted it into that site replacing the ? with each of the four options. In each case there was no corresponding sequence on record. That makes me think that this question is not worthwhile to work on. I cannot think of a time when I did not find at least one sequence which matched part of the sequence about which I asked.
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I would say, rather, that the sort of "pattern" they seem to like for the more complicated of these puzzles is not mathematically interesting, so it is not covered by OEIS. And that correlates well with the judgment that it is not worth the time.

Saumyojit said:
Actually yes, the pattern is consistent in the sense that when we are traveling from the odd term to even term everytime we are adding and from even to odd we are subtracting.

But the Syntax is same but addition and subtraction Will alternatively put into effect
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Yes, there is a "pattern" involved.

The trouble that such a "pattern", with successive terms determined by alternate operations with changing numbers, has more parameters than the number of terms given. What they think of as a "pattern" is not justified by the information provided. In particular, when you finally find something that yields an answer in the list, there is no reason to be sure that is what they intended, so there is no real feeling of satisfaction such as a real math problem produces; you just happened to get the same result someone else got.

Bottom line: I don't find it worth the time to try to find alternative "patterns" for different results.
 
I must agree, I find these puzzles a bit annoying; a feeling that is fed by the fact that I'm not 'good' at them - I can never guess what the person was thinking of. Other people seem to have an annoyingly great knack for doing so!
The recursive definition of the sequence according to the 'pattern' described by Saumyojit is:

[MATH]a_1=7\\ a_{n+1}=n(a_n-(-1)^n (n+1))[/MATH]
 
I am stuck with another number series problem


47 58 71 79 95 ?



A
108
B
107
C
105
D
109
 
That one's obvious ... except the answer is not in the list!

Do you see why it should be 100?

(Obviously, I will not put in any effort beyond that.)
 
Good Heavens.
I know the answer!!
Is the answer D 109? Do you add up the digits and add that to the number to get the next term?
 
Dr.Peterson said:
That one's obvious ... except the answer is not in the list!

Do you see why it should be 100?

(Obviously, I will not put in any effort beyond that.)
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No no .

There is only one pattern existing that will give only one answer out of those 4 options only.

The pattern by Which you are getting 100 is irrelevant in this context as they have already given the options.
 
Saumyojit said:
No no .

There is only one pattern existing that will give only one answer out of those 4 options only.

The pattern by Which you are getting 100 is irrelevant in this context as they have already given the options.
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No, my point is that the problem itself is irrelevant. These can be fun puzzles, but they should not be thought of as tests of mathematical skill. If anything, they test what is called "lateral thinking" (and perseverance).

Can you prove your claim, that there is only one possible "pattern" that produces one of those answers? Without a proof, you can't make such a claim mathematically. And I very much doubt that those who write these problems bother with such a proof. Why? Because they are not thinking mathematically. They are just asking you to guess how they got theirs.

I hope you are aware that the word "pattern" does not have a precise mathematical definition. That means that no proof of your claim is possible.
 
Dr.Peterson said:
No, my point is that the problem itself is irrelevant. These can be fun puzzles, but they should not be thought of as tests of mathematical skill. If anything, they test what is called "lateral thinking" (and perseverance).

Can you prove your claim, that there is only one possible "pattern" that produces one of those answers? Without a proof, you can't make such a claim mathematically. And I very much doubt that those who write these problems bother with such a proof. Why? Because they are not thinking mathematically. They are just asking you to guess how they got theirs.

I hope you are aware that the word "pattern" does not have a precise mathematical definition. That means that no proof of your claim is possible.
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@nasi112 gave the correct logic
It didn't came to my mind.
 
As I said, 10 or 15 minutes on excel will give you this. No irrational numbers can possibly come out of it. The 5! makes complete sense.

I do agree with Subhotosh Khan that no quintic is possible with integer coefficients. Whether that is true with polynomials of higher degree I do not know.

So @Saumyojit requested, here is a process that generates the sequence 7, 9, 12, 48, 190. 890.

[MATH]f(n) = \dfrac{1}{5!} * ( 338n^5 - 4865n^4 + 27320n^3 - 72655n^2 + 89462n - 38760).[/MATH]
Proof

[MATH]f(1) = \dfrac{338 - 4865 + 27320 - 72655 + 89462 - 38760}{120} = \dfrac{840}{120} = 7.[/MATH]
[MATH]f(2) = \dfrac{10816 - 77840 + 218560 - 290620 + 178924 - 38760}{120} =\\ \dfrac{1080}{120} = 9.[/MATH]
[MATH]f(3) = \dfrac{82134-394065+737640-653895+268386-38760}{120} =\\ \dfrac{1440}{120} = 12.[/MATH]
[MATH]f(4) = \dfrac{346112-1245440+1748480-1162480+357848-38760}{120} =\\ \dfrac{5760}{120} = 48.[/MATH]
[MATH]f(5) = \dfrac{1056250 - 3040625 + 3415000 - 1816375 + 447310 - 38760}{120} =\\ \dfrac{22800}{120} = 190.[/MATH]

[MATH]f(6) = \dfrac{2628288 - 6305040 + 5901120 - 2615580 + 536772 - 38760}{120} =\\ \dfrac{106800}{120} = 890.[/MATH]
There is no unique process, and these problems teach a bad lesson.
 
JeffM said:
So @Saumyojit requested, here is a process that generates the sequence 7, 9, 12, 48, 190. 890.

f(n)=15!∗(338n5−4865n4+27320n3−72655n2+89462n−38760).f(n)=15!∗(338n5−4865n4+27320n3−72655n2+89462n−38760).\displaystyle f(n) = \dfrac{1}{5!} * ( 338n^5 - 4865n^4 + 27320n^3 - 72655n^2 + 89462n - 38760).
Click to expand...

Yes I see there can be multiple Answers depending on the process how one thinks.
The answer is 172 given bcoz the question maker knows :
the formula that Jeff m gave, it Will not come to the mind of Students who will be solving this number Series rather the formula or pattern by Which we will be getting answer 172(shown some posts above ) that pattern can easily come to the student's mind.

You proved it but how did you came up with the formula.?

Genius ?
 
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Saumyojit said:
Yes I see .
You proved it but how did you came up with the formula.

Genius ?
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It is hardly genius. It just requires setting up a system of simultaneous equations and the knowledge that you can fit a unique polynomial of degree n to any n + 1 points. Solving a system of 6 simultaneous equations is mind numbing, but setting the equations up is not hard at all.

No, I am not any of those Jeff’s.
 
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