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Solution Techniques

$T(n) = \left\{ \begin{array}{ll} \Theta(1) & {\rm if} \; n = 0 \\ T(n-1) + \Theta(1) & {\rm if} \; n > 0 \end{array} \right.$

Guess: O(n)

Show: T(n) <= cn

Assume: T(n-1) <= c(n-1)

T(n) $\leq$ c(n-1) + $\Theta(1)$

= cn - c + $\Theta(1)$

$\leq$ cn

If c $\geq \Theta(1)$ . True for large enough c.

Initial Conditions: T(1) = $\Theta(1)$ $\leq$ cn

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