The number of substrings (of all length inclusive) that can be formed from a character string of length $$n$$ is

How many sub strings can be formed from a character string of length $$n$$?

Solve the recurrence equations:
$$\,\,\,\,\,\,\,\,\,\,T\left( n \right) = \left( {{n \over 2}} \right) + 1$$
$$\,\,\,\,\,\,\,\,\,\,\,T\left( 1 \right) = 1$$

(a) Solve the recurrence equations
$$\,\,\,\,\,\,\,\,\,T\left( n \right) = T\left( {n - 1} \right) + n$$
$$\,\,\,\,\,\,\,\,\,T\left( 1 \right) = 1T$$
(b) What is the generating function?
$$\,\,\,\,\,\,\,\,\,G\left( z \right)$$ for the sequence of Fibonacci numbers?