64, -48, 36, -27 which formula can be used to describe the sequence

A sequence is either arithmetic or geometric. The given sequence is geometric and the formula is [tex]-\frac{256}{3} (-0.75)^n[/tex]

First, we determine if the sequence is arithmetic or geometric.

An arithmetic sequence has a common difference (d) and it is calculated as follows:

[tex]d = T_2 - T_1[/tex] or [tex]d = T_3 - T_2[/tex]

So, we have:

[tex]d = -48 - 64 =-112[/tex]

[tex]d = 36--48 =84[/tex]

The 2 values of d are not equal; So, the sequence is not arithmetic.

Next, we check if it is geometric.

A geometric sequence has a common ratio (r) and it is calculated as follows:

[tex]r = \frac{T_2}{T_1}[/tex] or [tex]r = \frac{T_3}{T_2}[/tex]

So, we have:

[tex]r =\frac{-48}{64} = -0.75[/tex]

[tex]r =\frac{36}{-48} = -0.75[/tex]

The 2 values of r are equal; So, the sequence is geometric.

The formula of a geometric sequence is:

[tex]T_n = a \times r^{n-1}[/tex]

Where:

[tex]a = T_1=64[/tex] --- the first term

[tex]r = -0.75[/tex] --- the common ratio

So, we have:

[tex]T_n = a \times r^{n-1}[/tex]

[tex]T_n = 65 \times (-0.75)^{n-1}[/tex]

Apply law of indices

[tex]T_n = 64 \times \frac{(-0.75)^n}{-0.75}[/tex]

Rewrite as:

[tex]T_n = \frac{64}{-0.75}\times (-0.75)^n[/tex]

[tex]T_n = -\frac{64}{3/4}\times (-0.75)^n[/tex]

Apply reciprocals

[tex]T_n = -\frac{64\times 4}{3}\times (-0.75)^n[/tex]

[tex]T_n = -\frac{256}{3}\times (-0.75)^n[/tex]

[tex]T_n = -\frac{256}{3} (-0.75)^n[/tex]

Hence, the formula to describe the sequence is: [tex]-\frac{256}{3} (-0.75)^n[/tex]