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

Series
ANSWERED 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. 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