Recurrence Relation

Solve the recurrence relation$F_n=3F_{n-1}+10F_{n-2}+7.5^n$ where $F_0=4$ and $F_1=3$

Find the general solution of the recurrence relation: $a_n=a_{n-1}+6a_{n-2}$ with $a_0=3 \& a_1=6$

Find the general solution of the recurrence relation: $a_n=a_{n-1}+2a_{n-2}$ with $a_0=2 \& a_1=7$

Given that, $a_0=3$, solve the following recurrence relation:
$a_n=4-\frac{4}{a_{n-1}}$

Solve the recurrence relation $T\left(2^k\right)=3T\left(2^{k-1}\right)+1,T\left(1\right)=1$ 

Find the solution to the recurrence relation $a_n=6a_{n-1}+11a_{n-2}-6a_{n-3}$ given $a_0=20,a_1=5$ and $a_2=15$

What is a first order linear recurrence relation?

State the various types of recurrence relations.

Let $m, n$ be integers such that $1<m\le n.$ Define $f\left(m,n\right)=\left(1-\frac{1}{m}\right)\times \left(1-\frac{1}{m+1}\right)\times \left(1-\frac{1}{m+2}\right)\times \dots \times \left(1-\frac{1}{n}\right)$. 
If $S=f\left(2,2049\right)+f\left(3,2049\right)+f\left(4,2049\right)+\dots +f\left(2049,2049\right)$, find the value of $\frac{S}{64}$.

Define a function on the positive integers recursively by $f\left(1\right)=2 , f\left(n\right)=f(n-1)+1$$if n$ is even, and $f\left(n\right)=f(n-2)+2$ if $n$ is odd and greater than $1.$  What is $f\left(2017\right)-2000?$ 

The increasing sequence of positive integers $a_1, a_2, a_3, \dots ,$  has the property that $a_{n+2}=a_n+a_{n+1},$ for all $n\ge 1.$ Suppose that $a_7=120.$ What is $\frac{a_8}{2}?$

Suppose that the sequence $\left\{a_n\right\}$ is defined by $a_1=2,$ and $a_{n+1}=a_n+2nm,$ when $n\ge 1.$ What are last two digits of $a_{100} ?$

Let $a_n$ denote the number of all $n$ digit positive integers formed by the digits $0, 1$ or both such that no consecutive digits in them are $0.$ Let $b_n=$ the number of such $n$ digit integers ending with digit $1$ and $c_n=$ the number of such $n$ digit integers ending with digit $0.$

The value of $b_6$ is