how to find a formula for recursive succession

Question

I have to find a formula to calculate $a_n$ of this succession: $$ \begin{cases} a_{n+2}\equiv3a_{n+1}+10a_n\\ a_0 =2\\ a_1=3 \end{cases} $$ My question is: is there any technique for solving these types of exercises?

Answer 1

The famous technique is the characteristics equation, which for a recurrence of form $$\sum_{i=m_1}^{m_2}c_ia_{n+i}=0$$is defined as $$\sum_{i=m_1}^{m_2}c_i\lambda^i=0$$Let it has the roots $\lambda_1,\cdots ,\lambda_{m_2-m_1}$. Then the general formula for $a_n$ can be written as $$a_n=\sum_{i=1}^{m_2-m_1} K_iP_i(n)\lambda_i^n$$where the coefficients $K_i$ and the polynomials $P_i(n)$ of degree as much as the repetition of $\lambda_i$ can be obtained from initial conditions.

In this question, the characteristics equation is $$\lambda^2=3\lambda+10$$with roots $$\lambda_1=2\\\lambda_2=5$$hence $$a_n=K_1\cdot 2^n+K_2\cdot 5^n$$with $${K_1+K_2=2\\2K_1+5K_2=3}$$which yields $${K_1={7\over 3}\\K_2=-{1\over 3}}$$therefore $$ a_n={7\cdot 2^n-5^n\over 3} $$