The sequence is defined by a linear difference (or recursion) equation with constant coefficients. Such equations have solutions in the form of $r^n$ for some value(s) of $r$. Substituting such a "guess" into the equation results in

from which it follows that $r\=1\/2$ or $r\=1\/3$ and the general solution of the recursion equation is $a_n\=A\/2^n\+B\/3^n$. Applying the two initial conditions $a_1\=1$ and $a_2\=-1$ gives the two equations $A\/2\+B\/3\=1$ and $A\/4\+B\/9\=-1$, whose solution is $A\=-16\,B\=27$.

An explicit representation for the general term of the series is then $a_n\=27\/3^n-16\/2^n$, from which it is clear that the limit of the sequence $\left\{a_n\right\}$ is zero.$$

Table 8.1.8(a) contains the  task template that, given the general term of a sequence, calculates and graphs its first few members.