Recurrence relations

• A recurrence relation applies a function to the previous term to find the next term
• A sequence is converging if it moves towards a single value
• A sequence is diverging if it does not move towards a single value
• A sequence oscillates if the terms alternate between larger and smaller
• A sequence alternates if it switches between positive and negative terms
• A sequence is monotone if it only increases or decreases
• A sequence is strictly increasing/decreasing if each value is greater/less than (not equal to) the last
• A sequence is periodic if it has repeating sequences of numbers - the period is the number of terms in the repeating pattern
• To find the limit of a converging sequence, set $u_{n}$ = $u_{n+1}$ = L and solve the resulting equation
• A fibonacci type sequence is generated by adding together the two previous terms ($x_{n+2} =$$x_{n+1} +$$x_{n}$)
• The golden ratio ($\phi$) is equal to $\frac{1+\sqrt5}{2}$, and is the positive solution to $x^2 -x-1$. The negative solution is represented by $\phi_2$
• In general, the solution to $u_{n+1} =$$ku_{n}$ is $u_n =$$A \times k^n$, where A is a constant
• If $u_{n+1} =$$ku_n +$$f(n)$, the complementary solution is the solution to the relation ignoring $f(n)$, and the particular solution is found by guessing the an algebraic solution with a similar format to $f(n)$ and substituting into the relation to find the exact version
• If the guess for the particular solution is in the same form as the complementary solution, multiply the guess by n