SOLUTION: First number sequence: 75,15,25,5,15, ... The correct next number is 3 I don't get why? Second sequence: 183 305 527 749 961 ... The correct next number is 293. I a

Question 1195915: First number sequence:
75,15,25,5,15, ...
The correct next number is 3
I don't get why?

Second sequence:
183 305 527 749 961 ...
The correct next number is 293.
I also can't solve it.
Please help to understand how they both work?
Found 3 solutions by josgarithmetic, math_tutor2020, greenestamps:
Answer by josgarithmetic(39618)   (Show Source): You can put this solution on YOUR website!
A possibility for the first sequence, maybe; but I will just ignore the second sequence.

75, 15, 25, 5, 15, NEXT
3*5*5, 3*5, 5*5, 5 , 3*5

Starting from first term:
down, up, down, up, down.

Notice that the factors for each term made of combinations of 3 and 5, and none others. After the "15", the next term should go DOWN. The only way is for NEXT term to be 3.

Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

Question 1

For problems like this, we basically involve trial and error. There is no single approach. Effectively we have to get lucky to guess the pattern most of the time. With more practice, it should come easier.

The jump from 75 to 15 is "divide by 5" since
75/5 = 15

To get from 15 to 25, we add 10
15+10 = 25

Then from 25 to 5 is another "divide by 5" operation.
25/5 = 5

The pattern goes: "divide by 5, add 10, divide by 5, add 10" and so on.

Divide: 75/5 = 15
Add: 15+10 = 25
Divide: 25/5 = 5
Add: 5+10 = 15
Divide: 15/5 = 3

Keep in mind that sequence problems like this are fundamentally flawed.
Why is that?
Check out my previous response on this page for more information.
The summary of what I mention is that the sequence 1,2,4 could have infinitely many possible numbers as the fourth term; questions that ask about the next term are often too vague to answer. Further context is needed.

========================================================================================================

Question 2

This is another "think outside the box" type of question.
The approach is far from obvious.

The idea is to add 2 to each digit
183 breaks up into 1, 8, 3
Adding 2 to each gets:
1+2 = 3
8+2 = 10
3+2 = 5

So the 1,8,3 becomes 3,10,5
If we drop the tens digit "1" from "10", then we now have 3,0,5 hence the second term 305

Now let's repeat the process of adding 2 to each digit
3+2 = 5
0+2 = 2
5+2 = 7
It matches with the 527 as the third term. So far, so good.

Repeat again:
5+2 = 7
2+2 = 4
7+2 = 9
This matches as well.

And,
7+2 = 9
4+2 = 6
9+2 = 11
We drop the tens digit of "11" to be left with "1"
So we land on 961

Repeat one final time
9+2 = 11
6+2 = 8
1+2 = 3
We arrive at 183 (not 293)
The terms of the sequence will loop at this point since we arrived at one of the values previously mentioned.

Like with question 1, there are likely infinitely many ways we could generate the original sequence of {183, 305, 527, 749, 961}, which may or may not lead to 183 or 293 as the next term (or perhaps any number really).

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

There is no formal mathematics involved in finding the answer to ANY problems like these. As one of the tutors says in his response, finding AN answer is matter of trial and error, and of luck.

And it is ABSURD for the statement of the problem to say what the "correct" next numbers are. It is undoubtedly true that those next numbers are "correct" IF YOU USE THE SAME RULE AS THE PERSON WHO CREATED THE PROBLEM; however, in ANY such problem, ANY next number will form a valid sequence.

For the first example, the repeated rule "divide by 5; add 10" is VERY LIKELY the answer according to the creator of the problem. However, it CANNOT be said that the next number obtained using that rule is THE correct answer.

For the second example, one of the tutors uses an argument for how to find that the "correct" next number is 293; however, his rule is not completely consistent.

Using his basic idea, I see a different pattern, leading to a different next number:
The first digits of the given numbers are 1, 3, 5, 7, 9; so the first digit of the next number should again be 1.
The second digits of the given numbers are 8,0,2,4,6; so the second digit of the next number should again be 8.
The third digits of the given numbers are 3, 5, 7, 9, 1; so the third digit of the next number should again be 3.

So the next number should be 183, not 293.

So I obtain a next number that is different than the given "correct" answer, using a pattern that is more consistent than the pattern used by the other tutor to match the given "correct" answer.

So is my answer right, and the "correct" answer given in the statement of the problem is wrong?

Of course we can't say that. The problem -- like ALL other similar problems -- is fatally flawed; given ONLY a sequence of numbers without any context, it is ALWAYS IMPOSSIBLE to know what any other numbers in the sequence are.

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