Question O. Prove the closed form summation proposition P(n): a v(0
O. Prove the closed form summation proposition P(n): a v(0 = {1, 2, 3, .. Three steps are required: (a) first show the basis case and then (b) show that P(k) P(k + 1) is true; (c) prove and then claim the final result is true by the inductive hypothesis: VnP(n). Recall the axiom of induction is, in logical symbols, VP. [[P(0) A V(k E N). [P(k) P(k + 1)]] >(n e N). P(n)] where P is any predicate and k and n are both natural numbers. Number your three steps clearly. S(N)=a+a r+a r^{2}+a r^{3}+\cdots+a r^{N-1}=\sum_{k=0}^{N-1} a r^{k}=a \frac{1-r^{N}}{1-r}
= {1, 2, 3, .. Three steps are required: (a) first show the basis case and then (b) show that P(k) P(k + 1) is true; (c) prove and then claim the final result is true by the inductive hypothesis: VnP(n). Recall the axiom of induction is, in logical symbols, VP. [[P(0) A V(k E N). [P(k) P(k + 1)]] >(n e N). P(n)] where P is any predicate and k and n are both natural numbers. Number your three steps clearly. S(N)=a+a r+a r^{2}+a r^{3}+\cdots+a r^{N-1}=\sum_{k=0}^{N-1} a r^{k}=a \frac{1-r^{N}}{1-r}
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