Recurrence Equations Can someone explain how that Tn = 2Tn-1 + 1 sequence actually works? 
 A: The subscript merely indicates which term of the sequence we are using. Thus, $T_n$ represents the value of the nth term of the sequence.  The above interpretation applies equally well to any other term (eg. $n - 1$). 
$T_n = 2T_{n-1} + 1$ means "to find the value of the nth term of the sequence, you need to find the value of $(n - 1)$th term first; then 2 times it; and then add 1 to it".
A: The recurrence sequence is a sequence that each term is defined by its previous terms. Hence, the sequence must have the initial term. For example,
$T_n=2T_{n-1}+1.$ Assume that the sequence begin at $T_1.$
So, $T_2=2T_1+1$,
$T_3=2T_2+1=2(2T_1+1)+1=4T_1+3$ and so on.
We can see that if we do not know the value of $T_1$, we also can not find the value 
of $ T_2,T_3,...$ 
In fact, we can define recurrence relation that T_n is defined by T_{n-2}. For example,
$T_n=T_{n-2}+1$.
But in this case, the initial values must have 2 terms(T_1 and T_2).
If you want to find $T_{1000}$ of a recurrence sequence, you have to find $T_{999},T_{998},...$. It is not difficult, but it takes so much time. So, how can we find $T_{1000}$ without finding $T_{999}, T_{998},...$? That is the goal of recurrence equation.
A: A recurrence relation is a rule for generating a sequence of numbers (loosely). So let's say I give you a number, for instance $0$ and a rule: double your number and add 1. If you apply the rule repeatedly (starting with zero), you will get a sequence: $$(0,1,3,7,15,31,63,127...$$
More formally, we could say that $T_n$ is the $n$-th number in the sequence, starting from $n=0$. So for example, $T_0=0$, $T_1=1$ and $T_5=31$. Then the rule double your number and add 1 becomes: $$T_n=2T_{n-1}+1$$
Does this help?
