To do so we need to analyze how consecutive terms are related. Let's find their difference.
ccccc
a_2-a_1&=& 2-4 &=& - 2 [1.2em]
a_3-a_2&=& 5-2 &=& 3 [1.2em]
a_4-a_3&=& 1-5&=& - 4 [1.2em]
a_5-a_4&=& 6-1&=& 5 [2em]
... & & ... & & ...
a_n-a_(n-1) = - n*(-1)^n
We can see above that the difference between consecutive terms is - n*(-1)^(n-1). Therefore, to obtain the value of the term in the n^(th) position, we need to subtract - n*(-1)^n from the previous term. With this information and knowing that the first term equals 4, we can write the recursive formula.
a_1=4 and a_n=a_(n-1) - n*(-1)^n
