Provided a function that would be defined fo every natural number n as: g(n)=22+14+18+⋅⋅⋅+12n−1+12n 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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Provided a function that would be defined fo every natural number n as:  g(n)=22+14+18+⋅⋅⋅+12n−1+12n  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.

irwchh · Accepted Answer

Prove the following using the principle of mathematical induction: ∑j=1nj=12n−1+12n,for n&amp;gt;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&amp;gt;0. For each integer n, let P(n) be the statement  ∑j=1nj=21−n+2−n. Consider the following properties: base case: P(1) is true inductive step: For all integers k&amp;gt;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&amp;gt;0, the statement P(n) is true  Hint: Substitute n = 1 into the claim to verify P(1). Substitute n = 1 into ∑j=1nj=21−n+2−n: ∑j=11j=21−1+2−1  Hint: Simplify both sides. Simplify both sides to check for validity: Answer:   1≠32, therefore the base case is false and the proof is unsuccessful

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

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