Explain how to find the nth term of a harmonic sequence.

To find the nth term of a harmonic sequence, use the formula: a(n) = 1/n.

A harmonic sequence is a sequence of numbers where each term is the reciprocal of a positive integer. For example, the first few terms of a harmonic sequence are: 1, 1/2, 1/3, 1/4, 1/5, ...

To find the nth term of a harmonic sequence, use the formula: a(n) = 1/n. This means that the nth term is equal to one divided by n. For example, the 5th term of the harmonic sequence is a(5) = 1/5.

To prove this formula, we can use mathematical induction. First, we show that the formula holds for the first term: a(1) = 1/1 = 1. Next, we assume that the formula holds for some arbitrary value of n, and we want to show that it also holds for n+1.

So, assuming that a(n) = 1/n, we want to show that a(n+1) = 1/(n+1). We can do this by using the definition of a harmonic sequence: a(n+1) = 1/(n+1) if and only if the sum of the first n+1 terms is equal to the sum of the first n terms plus 1/(n+1).

We can prove this by using the formula for the sum of a harmonic sequence: S(n) = 1 + 1/2 + 1/3 + ... + 1/n. Then, the sum of the first n terms is S(n) = 1 + 1/2 + 1/3 + ... + 1/n, and the sum of the first n+1 terms is S(n+1) = 1 + 1/2 + 1/3 + ... + 1/n + 1/(n+1).

We can rewrite S(n+1) as S(n) + 1/(n+1), which shows that the sum of the first n+1 terms is equal to the sum of the first n terms plus 1/(n+1). Therefore, a(n+1) = 1/(n+1), and the formula holds for all positive integers n.

In summary, to find the nth term of a harmonic sequence, use the formula a(n)