Math CentralQuandaries & Queries


Question from Alan:

My son has the following sequence to work out the formula for the nth term (and find 5th and 6th terms) and I have tried a variety of ways but can't help!

1/2, 5/6, 1, 11/10

I have converted all to 15/30, 25/30, 30/30, 33/30 and so think the next two terms are 37/30 and 45/30 but I have no idea how to prove or if I am right!

Any help appreciated.

Hi Alan,

I did what you did and found a common denominator but could see no pattern in the fractions or the numerators. I then went to my favourite source for this type of problem, The On-Line Encyclopedia of Integer Sequences. I input the numerators, 15, 25, 30, 33 and got the reply "Not a prime and average of divisors except itself is an integer."

For example for 15 it is not a prime and its divisors are 1, 3, 5 and 15. Ignoring 15 as a divisor the others are 1, 3, and 5 and the average is 9/3 = 3 which is an integer so 15 is a member of the sequence. On the other hand 16 is also not a prime and its divisors are 1, 2, 4, 8 and 16. Ignoring 16 the other divisors have a mean of 15/4 which is not an integer so 16 is not a member of this sequence.

I think this involves way too much calculation to be the desired answer. My feeling is that there is a typo in the question.


Alan wrote back:

Thanks to Penny for trying to answer my earlier query re finding 5th and 6th numbers in a sequence, 1/2, 5/6, 1, 11/10 and an equation for the nth term. I’m pleased to say that I continued to play with it and converted as follows: 2/4, 5/6, 8/8, 11/10 and at last worked it out,

(3n-1)/(2n+2). So 5th and 6th terms are 14/12 and 17/14. Thought you might appreciate the answer and it wasn’t a typo! A very challenging question for a 13 year old I thought!

Thanks for this Alan. Nicely done!


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