Techniques for solving complex number series

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No Name

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Hello,

I'd like to ask if there are any special techniques one can apply to solve complex number series in 30 seconds or less. I've tried searching online, but the examples I find are fairly simple. Below are some examples of complex number series which I'm not sure how one should approach to solve quickly:

292 | 149 | 82 | 44 | 14 | ?

42 | 84 | 2 | 4/3 | 4/9 | ?

33 | 119 | 162 | 202 | 362 | 527 | ?

Please note that I already have the solutions, and when I go through them, it makes sense. But just looking at the series without the solutions...yeah... how is one supposed to figure out the relationship between adjacent values when the numbers don't appear to correlate in any meaningful way?

Surely there are techniques to solve such questions quickly?

Thanks in advance!
 
No Name said:
Hello,

I'd like to ask if there are any special techniques one can apply to solve complex number series in 30 seconds or less. I've tried searching online, but the examples I find are fairly simple. Below are some examples of complex number series which I'm not sure how one should approach to solve quickly:

292 | 149 | 82 | 44 | 14 | ?

42 | 84 | 2 | 4/3 | 4/9 | ?

33 | 119 | 162 | 202 | 362 | 527 | ?

Please note that I already have the solutions, and when I go through them, it makes sense. But just looking at the series without the solutions...yeah... how is one supposed to figure out the relationship between adjacent values when the numbers don't appear to correlate in any meaningful way?

Surely there are techniques to solve such questions quickly?

Thanks in advance!
Click to expand...
Not that I know of. Not without a computer....
 
The fact is that, given any finite list of numbers, the "next" number in the sequence can by anything! Such problems are really asking you to guess a simple rule. You have to assume there is such a rule!

Given any list of "n" numbers there exist a polynomial, in x, of degree n-1 that will give those numbers for x= 1 to n. "Newton's divided difference formula" can find that polynomial but it might not be the rule intended.

For example, the list 292, 149, 82, 44, 14 has "first differences" 149- 292= -143, 82- 149= -67, 44- 82= -38, and 14- 44= -30; "second differences" -67+ 143= 76, -38+ 67= 29, and -30+ 38= 8; "third differences" 29- 76= -47 and 8- 29= -21; "fourth difference" -21+ 47= 26 so we can write P(x)= 292- 143(x-1)+ (76/2)(x-1)(x-2)- (47/3!)(x-1)(x-2)(x-3)+ (26/4!)(x-1)(x-2)(x-3)(x-4). Then P(1)= 292, P(2)= 292- 143= 149, etc. Taking x= 5, the "next" term in the sequence, if this is the correct rule, is P(5)= 292- 143(4)+ 38(4)(3)- (47)(4)+ (13)(2)(1)= -38646 but we have no way of knowing if this is what was intended.
 
HallsofIvy said:
The fact is that, given any finite list of numbers, the "next" number in the sequence can by anything! Such problems are really asking you to guess a simple rule. You have to assume there is such a rule!

Given any list of "n" numbers there exist a polynomial, in x, of degree n-1 that will give those numbers for x= 1 to n. "Newton's divided difference formula" can find that polynomial but it might not be the rule intended.

For example, the list 292, 149, 82, 44, 14 has "first differences" 149- 292= -143, 82- 149= -67, 44- 82= -38, and 14- 44= -30; "second differences" -67+ 143= 76, -38+ 67= 29, and -30+ 38= 8; "third differences" 29- 76= -47 and 8- 29= -21; "fourth difference" -21+ 47= 26 so we can write P(x)= 292- 143(x-1)+ (76/2)(x-1)(x-2)- (47/3!)(x-1)(x-2)(x-3)+ (26/4!)(x-1)(x-2)(x-3)(x-4). Then P(1)= 292, P(2)= 292- 143= 149, etc. Taking x= 5, the "next" term in the sequence, if this is the correct rule, is P(5)= 292- 143(4)+ 38(4)(3)- (47)(4)+ (13)(2)(1)= -38646 but we have no way of knowing if this is what was intended.
Click to expand...
This method also assumes that the sequence is a polynomial.
 
HallsofIvy said:
The fact is that, given any finite list of numbers, the "next" number in the sequence can by anything! Such problems are really asking you to guess a simple rule. You have to assume there is such a rule!

Given any list of "n" numbers there exist a polynomial, in x, of degree n-1 that will give those numbers for x= 1 to n. "Newton's divided difference formula" can find that polynomial but it might not be the rule intended.

For example, the list 292, 149, 82, 44, 14 has "first differences" 149- 292= -143, 82- 149= -67, 44- 82= -38, and 14- 44= -30; "second differences" -67+ 143= 76, -38+ 67= 29, and -30+ 38= 8; "third differences" 29- 76= -47 and 8- 29= -21; "fourth difference" -21+ 47= 26 so we can write P(x)= 292- 143(x-1)+ (76/2)(x-1)(x-2)- (47/3!)(x-1)(x-2)(x-3)+ (26/4!)(x-1)(x-2)(x-3)(x-4). Then P(1)= 292, P(2)= 292- 143= 149, etc. Taking x= 5, the "next" term in the sequence, if this is the correct rule, is P(5)= 292- 143(4)+ 38(4)(3)- (47)(4)+ (13)(2)(1)= -38646 but we have no way of knowing if this is what was intended.
Click to expand...

Thanks for the explanation. Probably should have posted this earlier. For the first example, here is a snapshot of the solution the question intended to receive:

1576080788385.png

Even so, it really does seem like guessing would be the only way to go, since the other two examples have completely different solutions.
 
Subhotosh Khan said:
This method also assumes that the sequence is a polynomial.
Click to expand...
Since you are never given any real constraints on these type problems then there should not be any problem using a polynomial.

I like the following: On an exam put down any number for the next number, then go home and find a polynomial that works. When the teacher marks it wrong.... well you get where I am going. It also frees up time to work on other problems if you do not see the the obvious number immediately
 
No Name said:
Thanks for the explanation. Probably should have posted this earlier. For the first example, here is a snapshot of the solution the question intended to receive:

View attachment 15378

Even so, it really does seem like guessing would be the only way to go, since the other two examples have completely different solutions.
Click to expand...
When I see "292 | 149 | 82 | 44 | 14 | ? " I assume the rules applies to all numbers (except for the first), not pairs. Is this a thing??? This would be pretty scary. Were the numbers shown like this: "292 | 149 | 82 | 44 | 14 | ?"?
 
lev888 said:
When I see "292 | 149 | 82 | 44 | 14 | ? " I assume the rules applies to all numbers (except for the first), not pairs. Is this a thing??? This would be pretty scary. Were the numbers shown like this: "292 | 149 | 82 | 44 | 14 | ?"?
Click to expand...

Yes, the numbers are written exactly like that. Also, here's the solutions for the other two examples:

2. 42 | 84 | 2 | 4/3 | 4/9 | ?

1576085488259.png


3. 33 | 119 | 162 | 202 | 362 | 527 | ?

1576085291867.png


These are from an online preparation package and intended to help prepare for a supervised public service entry exam. My concern is that the prep pack advises that such questions should be solved in 30 seconds (maybe a minute). The solutions all make sense, but I am at a loss on how one should approach this madness.
 
No Name said:
These are from an online preparation package and intended to help prepare for a supervised public service entry exam. My concern is that the prep pack advises that such questions should be solved in 30 seconds (maybe a minute). The solutions all make sense, but I am at a loss on how one should approach this madness.
Click to expand...
I think all you can do is to recognize it as madness. That is indeed what it is: Your first example is a function problem made to look like a sequence problem (because each pair is unrelated to the previous pair); these last two are entirely different from that, but somewhat similar to one another.

It appears that they are trying to reward "lateral thinking" and perhaps penalize logic. There is no method to solve problems whose answers are so diverse; just let your mind go wild. Knowing what sorts of patterns they like, you may want to look at differences and ratios among terms, but keep your mind very open in looking for possible patterns, not expecting consistency.

I wonder if I'd want to work for an organization that prioritized this sort of thinking.
 
Thank you for the input Dr. Peterson. At least I can take comfort in knowing that these questions are indeed psychotic. Might be best to focus in other areas of the exam.
 
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