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Provided a function that would be defined fo every natural number n as

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asked 2021-09-03
Provided a function that would be defined fo every natural number n as: \[g(n)=\frac{2}{2}+\frac{1}{4}+\frac{1}{8}+\cdot\cdot\cdot+\frac{1}{2^{n-1}}+\frac{1}{2^{n}}\] COnjecture a closed form form for g(n) (a simple expression devoid of signs or dots) while also using mathematical induction to prove that the conjecture is true.

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2021-09-04
Prove the following using the principle of mathematical induction: \[\sum_{j=1}^nj=\frac{1}{2^{n-1}}+\frac{1}{2^{n}}, \text{for}\ n>0\] Hint: Determine the base case value. Since the base case value must be greater than 0, the base case value will be set to the next integer after 0: The base case value is n = 1 Hint: Find a proposition P(n) to be proved for all n>0. For each integer n, let P(n) be the statement \[\sum_{j=1}^nj=2^{1-n}+2^{-n}\]. Consider the following properties: base case: P(1) is true inductive step: For all integers k>0, if P(k) is true, then P(k + 1) is true If the above properties hold, then for each n element Z where n>0, the statement P(n) is true Hint: Substitute n = 1 into the claim to verify P(1). Substitute n = 1 into \[\sum_{j=1}^nj=2^{1-n}+2^{-n}\]: \[\sum_{j=1}^1j=2^{1-1}+2^{-1}\] Hint: Simplify both sides. Simplify both sides to check for validity: Answer: \[1\neq\frac{3}{2}\], therefore the base case is false and the proof is unsuccessful
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