1. Prove that for each n in N, 1+2++ n = n(n+1)/2. 2. Prove that for each n in N, 13 +23+ 3. Prove that for each n in N, 1+3+5+1 4. Prove that for each n ≥ 4,2" -1, then (1+x)" ≥1+nx for each n in N. 11. Prove DeMoivre's Theorem: fort a real number, (cost+i sint)" = cos nt + i sinnt for each n in N, where i = √√-1.

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2. Sam and Deb have a weekly net income of $1500. They have a pet dog. Their monthly expenses, not related to housing, are $2875. They have savings of $32 000. They are considering two housing options: Option 1: Renting a 2-bedroom condo for $1650 a month, plus utilities averaging $210 a month Option 2: Buying a 2-bedroom condo for a down payment of $24 500, bi-weekly mortgage payments of $1100, and a monthly condo fee of $475 a) Determine the monthly cost of each housing option. Factoring in other expenses not related to housing, which one can Sam and Deb afford? b) Suppose their dog falls ill and they have to pay $85 every week to cover veterinarian and medical expenses. Calculate the additional monthly expenses. How much money would be available for savings if they choose housing option 2?