Pay close attention to how the consecutive terms are related.
Recursive Rule: a_1 = 4, a_2 = 3, a_n=a_(n-2) - a_(n-1) Next Three Terms: 7, - 11, 18
Practice makes perfect
We want to write a recursive formula for the given sequence.
4, 3, 1, 2, - 1, 3, - 4, ...
To do so we need to analyze how the consecutive terms are related. Let's find the difference between each pair of consecutive terms.
\begin{array}{ccccc}
a_1-{\color{#0000FF}{a_2}}&=& 4 - {\color{#0000FF}{3}} &=& {\color{#009600}{1}} \\[1.2em]
{\color{#0000FF}{a_2}}-{\color{#009600}{a_3}}&=& {\color{#0000FF}{3}}-{\color{#009600}{1}} &=& {\color{#FF0000}{2}} \\[1.2em]
{\color{#009600}{a_3}}-{\color{#FF0000}{a_4}}&=& {\color{#009600}{1}}-{\color{#FF0000}{2}}&=& {\color{#FF00FF}{\text{-} 1}} \\[1.2em]
{\color{#FF0000}{a_4}}-{\color{#FF00FF}{a_5}}&=& {\color{#FF0000}{2}}-({\color{#FF00FF}{\text{-} 1}})&=& {\color{#A800DD}{3}} \\[1.2em]
{\color{#FF00FF}{a_5}}-{\color{#A800DD}{a_6}}&=& {\color{#FF00FF}{\text{-} 1}}-{\color{#A800DD}{3}}&=& {\color{#0000FF}{\text{-} 4}} \\[2em]
\vdots & & \vdots & & \vdots
\end{array}
a_(n-2)-a_(n-1) = a_n
We can see above that the difference of the consecutive terms equals the next term. Therefore, to obtain the value of the term in the n^(th) position, we need to subtract two previous terms. With this information and knowing that the first term equals 4 and the second term equals 3, we can write the recursive formula.
a_1 = 4, a_2 = 3 and a_n=a_(n-2) - a_(n-1)
We will write the next three terms of the sequence now.
ccccc
... & & ... & & ... [1.2em]
a_6- a_7&=& 3-( - 4) &=& 7 [1.2em]
a_7- a_8&=& - 4- 7 &=& - 11 [1.2em]
a_8- a_9&=& 7-( - 11)&=& 18
