# If An is defined as a geometric sequence, is the sequence 1/an also a geometric sequence? Explain your reasoning

## Since An/An-1 = r

(1/An)/(1/An-1) = An-1/An = 1/r
so, the ratio between terms is still constant.

## No, the sequence 1/an is not necessarily a geometric sequence.

To understand this, let us first recall the definition of a geometric sequence. A geometric sequence is a sequence in which each term is found by multiplying the previous term by a constant value, known as the common ratio, denoted by 'r'.

In the case of the sequence An, it is given that it is a geometric sequence. Therefore, we can express it as:

An = A1 * r^(n-1),

where A1 is the first term and r is the common ratio.

Now, let's examine the sequence 1/an. If we express it in terms of the geometric sequence An, we have:

1/an = 1 / (A1 * r^(n-1)).

The above expression does not follow the form of a geometric sequence as it involves division by the terms of An, rather than multiplication. This means that the sequence 1/an does not have a common ratio between terms, which is a characteristic of a geometric sequence.

Hence, the sequence 1/an is not a geometric sequence.

## To determine if the sequence 1/An is also a geometric sequence when An is defined as a geometric sequence, we need to examine the properties of a geometric sequence.

A geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by a constant value called the common ratio (r). Mathematically, a geometric sequence is defined as:

An = a * r^(n-1)

Where An is the nth term, a is the first term, r is the common ratio, and n is the position of the term in the sequence.

Now, considering the sequence 1/An, let's find the pattern:

1/A1 = 1/(a * r^(1-1)) = 1/a
1/A2 = 1/(a * r^(2-1)) = 1/(a * r)
1/A3 = 1/(a * r^(3-1)) = 1/(a * r^2)
...

As we can see, the terms in the sequence 1/An do not follow a constant multiplication pattern. In a geometric sequence, each term is obtained by multiplying the previous term by the common ratio (r), and we don't have that in the sequence 1/An.

Therefore, the sequence 1/An is not a geometric sequence when An is defined as a geometric sequence because the terms of 1/An do not follow the multiplication pattern required for a geometric sequence.