Find the next term: 1/5, 2/8, 2/12, etc

Question

Find the next term in the following sequence:

$1/5$, $~~2/8$, $~~2/12$, $~~4/18$, $~~2/24$, $~~?$

It's my first question so it might come off as a bit too easy.

Probably would have made it harder by reducing the fractions as much as you could. — Spencerkatty, Commented Oct 3, 2015 at 18:03
@Spencerkatty Reducing in simplest form would have given multiple choices for each term .... it will also make it much tougher to go in the right direction ... — Sudhanshu, Commented Oct 3, 2015 at 23:12

Gamow · Accepted Answer · 2015-10-03 17:14:39Z

4

The given number sequence is derived from

the sequence of prime numbers:

$~~~~$ $2$, $~3$, $~5$, $~7$, $~11$, $~13$, $~17$, $~19$, $~23$, $~\ldots$

The two entries of every pair are

the difference and the sum of consecutive prime numbers

This yields

3-2/3+2, $~$5-3/5+3, $~$7-5/7+5, $~$11-7/11+7, $~$13-11/13+11, $~$17-13/17+13,

so that the next term will be $~$4/30.

answered Oct 3, 2015 at 17:14

Gamow

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$\begingroup$ You cracked it!! $\endgroup$
– Sudhanshu
Commented Oct 3, 2015 at 17:20
1

$\begingroup$ What was the inspiration behind the answer? $\endgroup$
– Sudhanshu
Commented Oct 3, 2015 at 17:21
$\begingroup$ He may have searched the series of numerators on OEIS. $\endgroup$
– ghosts_in_the_code
Commented Oct 4, 2015 at 10:33

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ghosts_in_the_code · Accepted Answer · 2015-10-03 16:02:23Z

1

Just a guess..

$\frac{2}{28}$

because

Sum of numerator and denominator of a term equals difference between numerator and denominator of the next term.

This probably implies that

The next term is of the type $\frac{a}{a+26}$

but I don't know why

I'm guessing $a=2$.

A more complete answer will be required.

answered Oct 3, 2015 at 16:02

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$\begingroup$ No. That is not the answer. Going by this method can give numerous answer. $\endgroup$
– Sudhanshu
Commented Oct 3, 2015 at 16:34
$\begingroup$ The actual logic gives one clear answer. $\endgroup$
– Sudhanshu
Commented Oct 3, 2015 at 16:35
$\begingroup$ He was right on the 26 bit, his reasoning is connected with how the numbers are calced. He just lacked the top number. By this I mean his answer is 'incomplete' rather then 'based on a wrong method'. $\endgroup$
– Tim Couwelier
Commented Oct 3, 2015 at 19:22

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2 Answers 2