Conside the following two functions:
$${g_1}(n) = \left\{ {\matrix{ {{n^3}\,for\,0 \le n < 10,000} \cr {{n^2}\,for\,n \ge 10,000} \cr } } \right.$$
$${g_2}(n) = \left\{ {\matrix{ {n\,for\,0 \le n \le 100} \cr {{n^3}\,for\,n > 100} \cr } } \right.$$ Which of the following is true:

$$\sum\limits_{1 \le k \le n} {O(n)} $$ where O(n) stands for order n is:

Express T(n) in terms of the harmonic number H_n = $$\sum\limits_{t = 1}^n {1/i,n \ge 1} $$ where T(n) satisfies the recurrence relation, T(n) = $${{n + 1} \over 2}$$ T(n-1) + 1, for $$n \ge 2$$ and T(1) = 1 What is the the asymptotic behavior of T(n) as a function of n?

Solve the recurrence equations
T (n) = T (n - 1) + n
T (1) = 1