The required explicit formula is [tex]a_n = -3(a_{n - 1})[/tex]
Solution:
Given that sequence is -4, 12, -36, 108
To find: explicit formula
Explicit formulas define each term in a sequence directly, allowing one to calculate any term in the sequence
An explicit formula designates the nth term of the sequence, as an expression of n (where n = the term's location). It defines the sequence as a formula in terms of n.
Let us first find the logic used in sequence
[tex]\begin{array}{l}{-4 \times-3=12} \\\\ {12 \times-3=-36} \\\\ {-36 \times-3=108}\end{array}[/tex]
So we can see clearly that next term in sequence is obtained by multiplying -3 with previous term
This can be defined in terms of "n"
[tex]a_n = -3(a_{n - 1})[/tex]
Where [tex]a_n[/tex] represents the next terms location and [tex]a_{n-1}[/tex] represents previous term location
So the required explicit formula is [tex]a_n = -3(a_{n - 1})[/tex]
Let us verify our explicit formula
Now let us find the 4th term of sequence
[tex]\text {so } a_{n}=a_{4} \text { and } a_{n-1}=a_{4-1}=a_{3}[/tex]
[tex]a_{4}=-3\left(a_{3}\right)=-3(-36)=108[/tex]
Thus using the explicit formula, next terms in sequence can be found
