Lecture 04 Ticket

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Consider the following method that on input a positive integer $n$ sums the odd numbers from $1$ to $2n-1$:

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  Sum(n):
    total <- 0
    for k = 1 up to n do
      total <- total + 2 * k - 1
    endfor
    return total

Observe that for $n=1,2,3,4$, $\mathrm{S}\mathrm{u}\mathrm{m}(n)$ returns the values $1,4,9,16$. In each case, the value returned by $\mathrm{S}\mathrm{u}\mathrm{m}(n)$ is $n^2$. Use induction to argue that this formula always holds: for every positive integer $n$, the value returned by $\mathrm{S}\mathrm{u}\mathrm{m}(n)$ is $n^2$.

Solution

We will argue the following claim