Simplify to get 21 = 213+1 + 272+1 - 1 Just show the steps in the algebra Simplify further to get 21 = 27+2 -1 Again just show the steps in the algebra to get there. Now look carefully at your formula. 21 = 2"+2 - 1 Suppose you just replace n with n+1 in the original formula. What do you get? Can you see that we have built up the n+1 formula from the n formula?Have we proven the n+1 formula works? Not quite yet. Let's summarize. . We supposed the n formula is correct. We then built up the n+1 formula by just adding 2" to each side. So ... We now know that the formula is correct for if the formula is correct for But that's not worth much if we do not know that the n formula is correct. Going back to our example from the presentation, we want to know whether lab is on today. We use the timetable to prove lab is on if it's Tuesday. Great, but we only know lab is on today if we know it is Tuesday. But now the base case (n=0) helps us. We already showed the formula worlds for n = 0. We know the formula works for n = (base case) because we But if it works for n =0 then it works for n+1 = But if it works for then it works for And so on. So now what have we proved?\fSo we now know the formula works for n = 0, 1, 2. That's a hint that it might work for all n greater than or equal to 0, but it is NOT a proof that it does. Let's suppose the formula works for some value of n. We won't specify what this value is so long as it is at least 0. Let's call this the n formula. Next we suppose the n formula is correct. Pi 2' = 210- - 1