Question

Find the explicit formula for the given sequence.
-4, 12, -36, 108

Answer 1

The required explicit formula is
`a_n = -3(a_(n - 1))`

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

`\begin{array}{l}{-4 *-3=12} \\\\ {12 *-3=-36} \\\\ {-36 *-3=108}\end{array}`

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"

`a_n = -3(a_(n - 1))`

Where
`a_n` represents the next terms location and
`a_(n-1)` represents previous term location

So the required explicit formula is
`a_n = -3(a_(n - 1))`

Let us verify our explicit formula

Now let us find the 4th term of sequence

`\text {so } a_(n)=a_(4) \text { and } a_(n-1)=a_(4-1)=a_(3)`

`a_(4)=-3\left(a_(3)\right)=-3(-36)=108`

Thus using the explicit formula, next terms in sequence can be found