Question
Let $r:$ Nat $\rightarrow$ Nat, where $r$ is defined by:$$\begin{gathered}r(0)=0, \\\text { and } \\r(k+1)=r(k)+2 k+3 .\end{gathered}$$Compute the first five values of this sequence, and then prove that, for any $n, r(n)=(n+1)^2-1$.
Step 1
We will use mathematical induction to prove this statement. Base case: $n = 0$ $r(0) = 0 = (0+1)^2 - 1 = 0$ Inductive step: Assume that for some $k$, $r(k) = (k+1)^2 - 1$. We want to prove that $r(k+1) = (k+2)^2 - 1$. $r(k+1) = r(k) + 2k + 3$ (by the Show more…
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