# SOLUTION: Use mathematical induction to prove the following: For each natural number n,1+5+9+....(4n-3)=n(2n-1).

Algebra ->  Algebra  -> Conjunction -> SOLUTION: Use mathematical induction to prove the following: For each natural number n,1+5+9+....(4n-3)=n(2n-1).      Log On

 Ad: Algebrator™ solves your algebra problems and provides step-by-step explanations! Ad: Algebra Solved!™: algebra software solves algebra homework problems with step-by-step help!

 Logic: Propositions, Conjunction, Disjunction, Implication Solvers Lessons Answers archive Quiz In Depth

 Question 495743: Use mathematical induction to prove the following: For each natural number n,1+5+9+....(4n-3)=n(2n-1).Answer by Edwin McCravy(8879)   (Show Source): You can put this solution on YOUR website!```1+5+9+...+(4n-3) = n(2n-1) Note: The last term is (4n-3) ------------------- It is only necessary to show the formula works for n=1 before showing that if it works for n = k it works for n = k+1 but I will show it works for 1, 2 and 3. It works for n=1 because Last term = 4n-3 = 4*1-3 = 4-3 = 1 1 = 1(2*1-1) 1 = 1(2-1) 1 = 1(1) 1 = 1 It works for n=2 because Last term = 4n-3 = 4*2-3 = 8-3 = 5 1+5 = 2(2*2-1) 1+5 = 2(4-1) 1+5 = 2(3) 1+5 = 6 It works for n=3 because Last term = 4n-3 = 4*3-3 = 12-3 = 9 1+5+9 = 3(2*3-1) 1+5+9 = 3(6-1) 1+5+9 = 3(5) 1+5+9 = 15 Now we assume that k is some integer for which the formula works (and we have 3 such values already for which we know the formula works.) So we assume 1+5+9+....(4k-3) = k(2k-1). What we want to show is that if we substitute k+1 for n in 1+5+9+...(4n-3) = n(2n-1) It will also work. That is we want to show that based on the assumption that it works for k, that we'll end up with this without the question mark over the equal sign: 1+5+9+...+[4(k+1)-3] ≟ (k+1)[2(k+1)-1] The question mark above the equal sign is to show that we have not shown that yet. The above is what we must show. We will simplify what we are to show: 1+5+9+...+[4k+4-3] ≟ (k+1)[2k+2-1] 1+5+9+...+(4k+1) ≟ (k+1)(2k+1) 1+5+9+...+(4k+1) ≟ 2kČ+3k+1 Now we will show it. We start with: 1+5+9+...+(4k-3) = k(2k-1) We add the k+1st term, which is (4k+1), to both sides, and hope that the right side comes out to to the expression above, which is 2kČ+3k+1: 1+5+9+...+(4k-3)+(4k+1) = k(2k-1)+(4k+1) 1+5+9+...+(4k-3)+(4k+1) = 2kČ-k+4k+1 1+5+9+...+(4k-3)+(4k+1) = 2kČ+3k+1 So we have shown that if the formula works for some value of n, say n=k, then it will always work for the next integer n=k+1. Therefore Since we have shown it works for n=k=1, it works for n=k+1=2 Since it works for n=k+1=2, it works for n=k+2=3 Since it works for n=k+2=3, it works for n=k+3=4 Since it works for n=k+3=4, it works for n=k+4=5 etc. etc. forever Edwin```