Question 1208497
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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, ...

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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, ... 

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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, ...

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Here is another possibility:

For the 1st term, n=1 
(a) substitute n=1 in the quadratic 11n<sup>2</sup>-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 11n<sup>2</sup>-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 11n<sup>2</sup>-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 11n<sup>2</sup>-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</pre>