Find the 10th and nth terms of the sequence 4,30/7,60/13,...

If we'll write the sequence of
reciprocals:

1/4, 7/30, 13/60,
....

and we'll calculate the difference between consecutive
terms:

7/30 - 1/4 =
-1/60

13/60 - 7/30 =
-1/60

..................................

we'll
notice that we'll obtain the common difference d = -1/60, so the sequence of reciprocal
terms is an a.s.

So, the given sequence is a harmonic
sequence.

Now, we'll calculate  a10 for the arithmetic
sequence:

a10 = a1 + 9d

a10 =
1/4 + 9*(-1/60)

a10 = 1/4 -
3/20

a10 = (5-3)/20

a10 =
1/10

So, the 10-th term of the H.P. is t10 =
10.

The n-th of the A.P.
is:

an = a1 + (n-1)d

an = 1/4
+ (n-1)(-1/60)

an = (15 - n +
1)/60

an =
(16-n)/60

The n-th term of the H.P. is tn =
60/(16-n).