To find an explicit formula for the nth term of the given sequence (125, -25, 5, ...), we need to identify the type of sequence we are dealing with. The sequence provided is:
125, -25, 5, ...
Firstly, notice how each term is generated from the previous one. If we take any term after the first and divide it by the term before it, we see a consistent pattern:
-25 / 125 = -1/5
5 / -25 = -1/5
This consistent ratio between consecutive terms indicates that we are dealing with a geometric sequence. A geometric sequence is defined by its first term (a_1) and a common ratio (r), and its nth term (a_n) is given by the formula:
a_n = a_1 * r^(n-1)
where
- a_1 is the first term of the sequence,
- r is the common ratio, and
- n is the term number.
From the sequence you provided:
- The first term (a_1) is 125,
- The common ratio (r) is -1/5 (since each term is -1/5 times the previous term).
Therefore, we can use these values to write the explicit formula for the nth term of the sequence:
a_n = 125 * (-1/5)^(n-1)
This formula allows us to calculate the value of any term in the sequence by simply plugging in the value of n (the term number we want to find).
