Find the 6th term of the harmonic sequence whose 1st term is 3 and 3rd term is 3/5

Question

Harmonic sequence

Answer 1

An arithmetic sequence (progression) is of the form a, a+d, a+2d, ..., where a is the first term and d the common difference between terms. The corresponding harmonic sequence is 1/a, 1/(a+d), 1/(a+2d), ...

If 1/a is 3 then a=1/3; 1/(a+2d)=3/5, so 3/5=1/(1/3+2d). 1/3+2d=5/3, 1+6d=5, 6d=4, d=2/3.

6th term=1/(a+5d)=1/(1/3+10/3)=3/11. (AP=1/3, 1, 5/3, 7/3, 3, 11/3, 13/3, ...)