Harmonic Sequence

A harmonic sequence is a sequence of numbers in which the reciprocal of each term is in arithmetic progression. In other words, the difference between the reciprocals of consecutive terms is constant.

Given Information

In this problem, we are given two terms of a harmonic sequence: a3 = 1 and a6 = 2.

Finding the First Term

To find the first term of the harmonic sequence, we need to determine the constant difference between the reciprocals of consecutive terms.

Finding the Common Difference

First, let's find the reciprocal of the third term, a3: 1/a3 = 1/1 = 1.

Next, let's find the reciprocal of the sixth term, a6: 1/a6 = 1/2.

Now, we can find the common difference between the reciprocals of consecutive terms: (1/2) - 1 = -1/2.

Finding the First Term

To find the first term, we need to find the reciprocal of the common difference: 1/(−1/2) = -2.

Therefore, the first term of the harmonic sequence is -2.

Summary

In summary, to find the first term of a harmonic sequence given two terms, we need to find the common difference between the reciprocals of consecutive terms. By subtracting the reciprocal of the sixth term from the reciprocal of the third term, we found that the common difference is -1/2. Taking the reciprocal of the common difference, we determined that the first term of the harmonic sequence is -2.