Question

11: I can identify the predicate being used in a proof by mathematical induction and use it to set up a framework of assumptions and conclusions for an induction proof. Below are three statements that can be proven by induction. You do not need to prove these statements! For each one clearly state the predicate involved; state what you would need to prove in the base case; clearly state the induction hypothesis in terms of the language of the proposition (i.e., without using notation to represent the predicate); and then clearly state the inductive step in terms of the language of the proposition. 1. For all positive integers n, 3+7+11 + ... + (4n - 1) = (4n+1)-1 2. Consider the set s defined recursively as follows. First we declare that 2 € S. Then for all E Sa+7 € S. The set S is equal to the set of positive integers k such that 7 divides k-2: that is, 7(K-2). 3. For all positive integers n. 2-0 (1) = 2".

Answer 1

Date Page 3 + 7 + 11 + This an a number + Tấn - ) is. Arithmetic progressien hewing common difference, d=4 sum of AR = atine (a+l) whee is first team and nis of teams , l is last team so, 3+7+11 + + 140-1) = n /3+ 40-1) 212+ 4n) 12)(1+2n) n(1+2n) =an² tn Now Rus is Rus is lunt} = 16n? +1+8n-1 2 Eln 8 =16n?ton 8 = 8(2n²+n) & 2n²tn 8 Plano 12 gn Set s. first team = 2 Next teams come up up by adding Hence it'll out to be an Arithmetic professiona ili cs fense, 916, 23:30, come CamScanner

Page SA be difference between each consecutive teom 7. se difference between any the foom will be multiple 7. Hence 7 diuides k-2 [ because is first tem & k can be any team in s. must multiple of S OR also see it like let k is nth term anik = a + (n-1) (d) =d+In-117)33+ n-7 = 7n-5 Noin kr 7-5-2 = 7n-7= 7(n-1) which ایل can is multiple of 7 3 (1) 3 lo 6. This to represents I izo TS notation combination t + € t + sy 0 ಇದರ it Meniu w n"+ <LD"10" + 0.01) +-- (1+1) es fared wis Busing Binomial Expansion Scanner