Question 983173
Find the 9th term of the harmonic sequence
{{{3/2}}},{{{1/2}}},{{{3/10}}},{{{3/14}}},... n=9
<pre>
Get their reciprocals:

{{{2/3}}},{{{2/1}}},{{{10/3}}},{{{14/3}}},... n=9

If we write the {{{2/1}}} with a 3 denominator, as {{{6/3}}}

we can see the arithmetic sequence which the reciprocals form:

{{{2/3}}},{{{6/3}}},{{{10/3}}},{{{14/3}}},... n=9

Extend it to 9 terms by adding common ratio {{{4/3}}} each time:

{{{2/3}}},{{{6/3}}},{{{10/3}}},{{{14/3}}},{{{18/3}}},{{{22/3}}},{{{26/3}}},{{{30/3}}},{{{34/3}}}

Reduce all the fraction that will reduce:

{{{2/3}}},{{{2/1}}},{{{10/3}}},{{{14/3}}},{{{6/1}}},{{{22/3}}},{{{26/3}}},{{{10/1}}},{{{34/3}}}

Take the reciprocals:

{{{3/2}}},{{{1/2}}},{{{3/10}}},{{{3/14}}},{{{1/6}}},{{{3/22}}},{{{3/26}}},{{{1/10}}},{{{3/34}}}

9th term is {{{3/34}}}

Edwin</pre>