Pay close attention to how the consecutive terms are related.
a_1=1, a_2=4, a_n=a_(n-2) + a_(n-1)
Practice makes perfect
We want to write a recursive rule for the given sequence. To do that, we first need to identify whether the given sequence is arithmetic, geometric, or neither. To do so we will calculate the difference and ratio between consecutive terms.
We can see above that neither the ratios nor the differences are the same. Therefore, the sequence is neither geometric nor arithmetic, so we have to look for another pattern. To do so we need to consider how the consecutive terms are related. Let's take a look at the differences between pairs of consecutive terms.
ccccc
a_1&=& 1 [1.2em]
a_2&=& 4 [1.2em]
a_3-a_2&=& 5-4 &=& 1 [1.2em]
a_4-a_3&=& 9-5&=& 4 [1.2em]
a_5-a_4&=& 14-9&=& 5 [2em]
... & & ... & & ...
a_n-a_(n-1) = a_(n-2)
We can see above that the difference between the consecutive terms equals a_(n-2). Therefore, to obtain the value of the term in the n^(th) position, we need to add two previous terms a_(n-2) and a_(n-1). With this information and knowing that the first and second term equal 1 and 4 we can write the recursive formula.
a_1=1, a_2=4, a_n=a_(n-2) + a_(n-1)
