Pay close attention to how the consecutive terms are related.
Recursive Rule: a_1 = 6, a_2 = 1, a_n=a_(n-1) + a_(n-2)
Next two terms: 38, 61
Practice makes perfect
We want to write a recursive formula for the given sequence.
6, 1, 7, 8, 15, 23, ...
To do so we need to analyze how the consecutive terms are related. Let's find the sum between each pair of consecutive terms.
\begin{array}{ccccc}
{\color{#0000FF}{a_2}}+a_1&=& {\color{#0000FF}{1}}+6 &=& {\color{#009600}{7}} \\[0.3em]
{\color{#009600}{a_3}}+{\color{#0000FF}{a_2}}&=& {\color{#009600}{7}}+{\color{#0000FF}{1}} &=& {\color{#FF0000}{8}} \\[0.3em]
{\color{#FF0000}{a_4}}+{\color{#009600}{a_3}}&=& {\color{#FF0000}{8}}+{\color{#009600}{7}}&=& {\color{#FF00FF}{15}} \\[0.3em]
{\color{#FF00FF}{a_5}}+{\color{#FF0000}{a_4}}&=& {\color{#FF00FF}{15}}+{\color{#FF0000}{8}}&=& {\color{#A800DD}{23}} \\[0.3em]
\vdots & & \vdots & & \vdots
\end{array}
a_(n-1)+a_(n-2) = a_n
We can see above that the sum of the consecutive terms equals the next term. Therefore, to obtain the value of the term in the nth position, we need to add two previous terms. With this information and knowing that the first term equals 6 and the second term equals 1, we can write the recursive formula.
a_1=6, a_2 = 1 and a_n=a_(n-1) + a_(n-2)
We will write the next two terms of a sequence now.
ccccc
... & & ... & & ... [0.3em]
a_6+ a_5&=& 23+ 15 &=& 38 [0.3em]
a_7+ a_6&=& 38+ 23&=& 61
