SOLUTION: 1/2, 2/3, 6/5,?

Question 1208497: 1/2, 2/3, 6/5,?
Found 4 solutions by greenestamps, Edwin McCravy, ikleyn, math_tutor2020:
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!

First fraction in the sequence: 1/2.

1*2 = 2; 1+2 = 3; next fraction in the sequence is 2/3.

2*3 = 6; 2+3 = 5; next fraction in the sequence is 6/5.

6*5 = 30; 6+5 = 11; next fraction in the sequence is 30/11.

ANSWER: 30/11

Note that this is only one possible answer based on a pattern that I saw. Other answers are possible based on patterns that other people might see.

So any "answer" to the problem is only a guess....

Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!

Below are four possibilities, which is to show you there are more than one possible answer when given a partial sequence to find the next term. Here is one possibility: You start with a fraction with the smallest positive integer as its numerator, and the smallest prime integer as its denominator. Then to get the next term, which is a fraction, multiply the numerator by the denominator to get the numerator of the next term, and use the next prime number as its denominator. You start with 1/2 as the first term. You multiply 1x2=2 to get the numerator of the 2nd term. You use the next prime 3 for a denominator. So the 2nd term is 2/3. You multiply 2x3=6 to get the numerator of the 3rd term. You use the next prime 5 for a denominator. So the 3rd term is 6/5. You multiply 6x5=30 to get the numerator of the 4th term. You use the next prime 7 for a denominator. So the 4th term is 30/7, 1/2, 2/3, 6/5, 30/7, ... ----------------------------------------------- Here is another possibility: The numerators are the factorials 1!=1, 2!=2, 3!=6, 4!=24 The denominators are the positive integers which are not divisible by any perfect square other than 1. Neither 2, 3, 5, are divisible by a perfect square other than 1, so the next one that isn't is 7. 1/2, 2/3, 6/5, 24/7, ... ---------------------------------- Here is another possibility: The numerators of each odd-numbered term is 1 more than a multiple of 5. The numerators of each even-numbered term is 1 more than the numerator of the preceding odd-numbered term. The denominator of each term is found by multiplying the number of term by one less, adding 4, then dividing by 2. The 1st term is odd-numbered because 1 is an odd number. So the numerator is 1 more than the 1st multiple of 5, which is 0. Add 1 and get 1. The denominator: 1x0=0, then add 4, get 4. Then divide by 2, get 2. So the 1st term is 1/2. The 2nd term is even-numbered because 2 is an even number. So the numerator is 1 more than the numerator of the preceding term, or 2 The denominator: 2x1=2, then add 4, get 6. Then divide by 2, get 3. So the 2nd term is 2/3 The 3rd term is odd-numbered because 3 is an odd number. So the numerator is 1 more than the 2nd multiple of 5, which is 5. Add 1 and get 6. The denominator: 3x2=6, then add 4, get 10. Then divide by 2, get 5. So the 1st term is 6/5. The 4th term is even-numbered because 4 is an even number. So the numerator is 1 more than the numerator of the preceding term, or 7 The denominator: 4x3=12, then add 4, get 16. Then divide by 2, get 8. So the 4th term is 7/8. 1/2, 2/3, 6/5, 7/8, ... --------------------------------- Here is another possibility: For the 1st term, n=1 (a) substitute n=1 in the quadratic 11n²-23n+42 (b) get 11-23+42=30 (c) divide by 60, get 30/60 which reduces to 1/3 For the 2nd term, n=2 (a) substitute n=2 in the quadratic 11n²-23n+42 (b) get 44-46+42=40 (c) divide by 60, get 40/60 which reduces to 2/3 For the 3rd term, n=3 (a) substitute n=3 in the quadratic 11n²-23n+42 (b) get 99-69+42=72 (c) divide by 60, get 72/60 which reduces to 6/5 For the 5th term, n=4 (a) substitute n=4 in the quadratic 11n²-23n+42 (b) get 176-92+42=126 (c) divide by 60, get 126/60 which reduces to 21/10 1/2, 2/3, 6/5, 21/10, ... Edwin

Answer by ikleyn(52788)   (Show Source): You can put this solution on YOUR website!
.

When a person comes to the forum with similar questions, it is a sign of mental degeneration.

This is a sign that a person does not understand fundamentally the essence of things in the nature.

The point is that the sequence of numbers itself does not determine "the next term".
In order for a sequence determines next term, an additional condition should be imposed, which is not presented in this problem.

When somebody gives you such a "problem", it means that this person either is mathematically illiterate
or wants to take your time and/or your money leaving nothing in exchange to you.

In other words, when someone gives you a problem like this,
keep an eye on your pockets so as not to end up a fool.

Such problems do not teach Mathematics, but on the contrary, corrupt the mind.

They are only good to raise demagogues.

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

There are no context clues or instructions telling us what type of sequence this is (whether it pertains to the entire fractions themselves or the pieces of the fractions).

Predicting the next term of a sequence like this is not really mathematics.
It's purely guesswork.
The other tutors have shown there are multiple possible answers.
There are perhaps infinitely many.

I'll provide a summary of what tutors Greenestamps and Edwin have found.
Each item in red is a potential answer, but there's no way of truly knowing without more context.
It might turn out that your teacher is looking for something else entirely.
1/2, 2/3, 6/5, 30/11
1/2, 2/3, 6/5, 30/7
1/2, 2/3, 6/5, 24/7
1/2, 2/3, 6/5, 7/8
1/2, 2/3, 6/5, 21/10
It would be cruel madness for a teacher to expect students to be mind readers.

See this similar question
On that link I try to find the next few terms in the sequence 1,2,4,...
It turns out there are at least 3 different possible answers for that question. There may be infinitely many answers.
This is more evidence that vague questions like this are very flawed. There needs to be more context provided.

A good well phrased question is something like "The sequence 1,3,5,7,... is arithmetic. What is the next term?"
A very flawed question simply gives a sequence without mentioning what type of sequence it is.

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