Determine if a sequence is harmonic.

To determine if a sequence is harmonic, we need to check if the reciprocals of the terms form an arithmetic sequence.

A sequence is said to be harmonic if the reciprocals of its terms form an arithmetic sequence. In other words, if the sequence is {a1, a2, a3, ...}, then it is harmonic if 1/a1, 1/a2, 1/a3, ... form an arithmetic sequence.

To check if a sequence is harmonic, we can calculate the difference between consecutive reciprocals and see if it is constant. If it is, then the sequence is harmonic. For example, consider the sequence {2, 4, 8, 16}. The reciprocals of these terms are {1/2, 1/4, 1/8, 1/16}. The differences between consecutive reciprocals are:

1/2 - 1/4 = 1/4
1/4 - 1/8 = 1/8
1/8 - 1/16 = 1/16

Since the differences are constant (1/4), the sequence is harmonic.

Another way to check if a sequence is harmonic is to calculate the sum of the reciprocals. If the sum is finite, then the sequence is harmonic. For example, consider the sequence {1, 2, 3, 4}. The reciprocals of these terms are {1/1, 1/2, 1/3, 1/4}. The sum of these reciprocals is:

1/1 + 1/2 + 1/3 + 1/4 = 2.0833...

Since the sum is finite, the sequence is harmonic.

In summary, to determine if a sequence is harmonic, we can check if the reciprocals of the terms form an arithmetic sequence by calculating the differences between consecutive reciprocals or by calculating the sum of the reciprocals.