Induction for each integer n 2 1, define the sum s(n) as follows i(i+1) 12 2 3 3-4 for example, s(1) s(2)=? and -i(i +1)-1-2 2 12 2.3 3 i. [2 points] compute each of the following values of s(n) and write your final answer as a fraction in lowest terms s(3) s(4) s(5)- ii. [1 point] do you see a general pattern for the simplified value of s(n) in general? write your guess to complete the proposition p(n) below for each integer n 2 1, let p(n) be the defined as the following proposition: iii. [4 points] prove that p(n) is true for all n 2 1 using a proof by induction. you must clearly state your induction hypothesis and indicate when it is used during the proof of your induction step. as usual, you must declare what each variable in your solution represents and make it clear whether each step of your proof is an assumption, something you are about to prove, or something that follows from a previous step or definition, etc

Submarines control how much they float or sink in the ocean by changing the volume of air and water contained in large ballast tanks. when the tanks are completely full of water, the submarine increases its overall mass and sinks down to the bottom. when the tanks are completely full of air, the submarine reduces its overall mass and floats to the surface. depending on the density of the seawater surrounding the submarine, it will pump seawater in or out of the tanks in order to achieve the same overall density as the sea water and float neutrally in the water. the volume of the submarine never changes. when the tanks are completely full of water, a submarine with a volume of 7.8\times10^3\text{ m}^37.8×10 3 m 3 7, point, 8, times, 10, start superscript, 3, end superscript, space, m, start superscript, 3, end superscript has a total mass of 8\times10^6\text{ kg}8×10 6 kg8, times, 10, start superscript, 6, end superscript, space, k, g. the density of the seawater is 10^3\text{ kg/m}^310 3 kg/m 3 10, start superscript, 3, end superscript, space, k, g, slash, m, start superscript, 3, end superscript. to make that submarine float neutrally, and neither float nor sink in the ocean, what volume of water does that submarine need to subtract from its tanks?