I have gone through the recurrence relation concept.
I have got to know two cases i.e.,
$1)$ $a_n = C^n a_{n-1}$
$2)$ $F_n = F_{n-1}-F_{n-2}$
But I got a question that
Find the explicit formula for the sequence defined by the recurrence relation
$a_n = 5\cdot a_{n-1} +3$ with initial condition $a_1 = 3$
I have tried to solve.But i could not match the above the recurrance relation to any of the First order or second order recurrence relation.
Can anyone help me to solve this.Please.
Best Answer
For your case, the standard approach is:
$1.$ Find a solution for $a_n=5a_{n-1}$ and,
$2.$ Find a particular solution for $a_n=5\cdot a_{n-1}+3$.
For the case $(1)$ we have geometric sequence which has a solution $h_n=a\cdot 5^n$
For the case $(2)$, the particular solution would be something related to last term $3$. That suggest us a constant solution $p_n=k$. If we replace that constant in the original recurrence we get:
$$k=5k+3 \Rightarrow k=-3/4$$
The general solution will be:
$$a_n=h_n+p_n=a\cdot 5^n-3/4$$
And using that
$$a_1=3= a\cdot 5^1-3/4 \Rightarrow a=3/4$$
and then
$$a_n=\frac{3}{4}\left(5^n-1\right)$$