Question

What 7th term of the harmonic sequence whos 3rd term is 4/3 and 5th term is 2/3

Answer 1

In a general harmonic sequence, each term is the reciprocal of a term in an arithmetic series.
e.g. if we have the AS
5,7,9,11, ...
then 1/5, 1/7, 1/9, 1/11 would be a harmonic sequence

so if 4/3 is the third term of the harmonic, then 3/4 is the third term of the AS
if 2/3 is the fifth term of the harmonic, then 3/2 is the fifth term of the AS
so let's find out about the AS
term(3) = a + 2d = 3/4
term(5) = a + 4d = 3/2
subtract them:
2d = 3/2-3/4 = 3/4
d = 3/8, then a = 0

then term(7) of the AS is a+6d = 0 + 6(3/8) = 9/4

and our 7th term of the Harmonic would be 4/9

check:
our AS would be
0, 3/8, 6/8, 9/8, 12/8, 15/8, 18/8, 21/8, ....
0, 3/8, 3/4, 9/8, 4/3, 15/8, 9/4, 21/8, ...

so term(7) of the AS is 9/4
and the term(7) of the harmonic would be 4/9

Answer 2

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