The value of each term of the sequence is given by the y-coordinates of the given points. Pay close attention to how consecutive terms are related.
Recursive Rule: a_1 = 64, a_2 = 16, a_n= a_(n-2)/a_(n-1) Next two terms: 4, 1/4
Practice makes perfect
We want to write a recursive formula for the sequence formed by the y-coordinates of the given points.
64, 16, 4, 4, 1 ...
To do so we need to analyze how the consecutive terms are related. Let's find the quotient of each pair of consecutive terms.
\begin{array}{ccccc}
\dfrac{a_1}{{\color{#0000FF}{a_2}}}&=& \dfrac{64}{{\color{#0000FF}{16}}} &=& {\color{#009600}{4}} \\[1.2em]
\dfrac{{\color{#0000FF}{a_2}}}{{\color{#009600}{a_3}}}&=& \dfrac{{\color{#0000FF}{16}}}{{\color{#009600}{4}}} &=& {\color{#FF0000}{4}} \\[1.2em]
\dfrac{{\color{#009600}{a_3}}}{{\color{#FF0000}{a_4}}}&=& \dfrac{{\color{#009600}{4}}}{{\color{#FF0000}{4}}}&=& {\color{#FF00FF}{1}} \\[1.2em]
\vdots & & \vdots & & \vdots
\end{array}
a_(n-2)/a_(n-1) = a_n
We can see above that the quotient of two consecutive terms equals the next term. Therefore, to obtain the value of the term in the nth position, we need to divide the previous two terms. With this information, and knowing that the first and second terms are 64 and 16, we can write the recursive formula.
a_1 = 64, a_2 = 16 and a_n=a_(n-2)/a_(n-1)
We will write the next two terms of a sequence now.
ccccccc
... & & ... & & ... & & ... [1.2em]
a_6 & = & a_4/a_5&=& 4/1 &=& 4 [1.2em]
a_7 & = & a_5/a_6&=& 1/4 &=& 1/4
