Find an explicit rule for the nth term of the sequence.
{eq}3,\ -12,\ 48,\ -192,\ \cdots {/eq}
Question:
Find an explicit rule for the nth term of the sequence.
{eq}3,\ -12,\ 48,\ -192,\ \cdots {/eq}
Geometric Sequence:
If we are given a sequence having terms that are produced by multiplying previous terms by a constant, then its general term can be generated by following the formula below:
{eq}a_n = a_1 r^{n-1} {/eq}
where
{eq}a_1 {/eq} is the first term
{eq}r {/eq} is a constant found by taking the quotient of any pair of consecutive terms.
Answer and Explanation:
The first term is {eq}a_1= 3 {/eq}.
The common ratio is {eq}r= \displaystyle \frac{-12}{3}=-4 {/eq}
The explicit rule for the {eq}n^{\mathrm{th}} {/eq} term is found by using {eq}a_n = a_1 r^{n-1} {/eq} and the values provided:
{eq}\begin{align*} \displaystyle a_n &= a_1 r^{n-1}\\\\ & =(3)(-4)^{n-1} \\\\ & =\bf{- \frac{3}{4}(-4)^{n }} \\\\ \end{align*} {/eq}
Learn more about this topic:
from
Chapter 21 / Lesson 9Get the geometric sequence definition and view examples. Learn how to find the nth term of a geometric sequence using the geometric sequence formula.