Question

Write an explicit formula for a, start subscript, n, end subscripta

n
​
, the n, start superscript, th, end superscriptn
th
term of the sequence 64, comma, minus, 16, comma, 4, comma, point, point, .64,−16,4,....

Answer 1

The explicit formula for the n-th term an of the given geometric sequence 64, -16, 4, ... with a common ratio of -1/4 is
`\[ a_n = 64 \cdot \left(-(1)/(4)\right)^((n-1)) \]`.

The given sequence is a geometric sequence with a common ratio of -1/4. In a geometric sequence, each term is obtained by multiplying the previous term by a fixed ratio. The explicit formula for the n-th term (a_n) of a geometric sequence is given by:

`\[ a_n = a_1 \cdot r^((n-1)) \]`

where:

(a_n) is the n-th term,

(a_1) is the first term,

(r) is the common ratio, and

(n) is the term number.

In this specific sequence, (a_1 = 64) and (r = -1/4). Substituting these values into the formula, we get the explicit formula for the n-th term:

`\[ a_n = 64 \cdot \left(-(1)/(4)\right)^((n-1)) \]`

This formula can be used to find any term in the sequence by substituting the desired value of (n). The sequence you provided is a geometric progression where each term is one-fourth of the preceding term, and this formula captures that pattern.

Complete question:

Write an explicit formula for a, start subscript, n, end subscripta n, the n, start superscript, th, end superscriptn th term of the sequence 64, comma, minus, 16, comma, 4, comma, point, point, .64,−16,4,....