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

$\begin{array}{l}{-4 \times-3=12} \\\\ {12 \times-3=-36} \\\\ {-36 \times-3=108}\end{array}$

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