(a) $3,6,9,10,12,13,15,16,18,19,20,21,22, \ldots$(b) A postage that $n$ cents deserve to be developed using just 3 -cent and also 10 -cent stampswith $n \geq 18$ .(c) A postage that $n$ cents deserve to be developed using just 3 -cent and also 10 -cent stampswith $n \geq 18$ .




You are watching: Determine which amounts of postage can be formed using just 3-cent and 10-cent stamps.

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Video Transcript

in component a to be asked to turn The postage is that might be created using only three cent and 10 cent stamps, therefore we can only use three cent in 10 cent stamps. Then, of course, we conform re feeling of postage 2 times 3 a 6 cents of postage attempts. 3 is rash of postage, 10 cent of postage, and also we have four temps. 3 is 12 cent of postage and 10 plus three is 13 cent a postage. 12 plus 3 is 15 feeling of postage, 13 plus three x 16 cent of postage, 15 plus three is 18 cent of postage and 16 plus three is 19 cents Postage in two times 10 is 20 cent postage. And also since we have there, we conform 18 19 and also 20 cents of postage. Us should have the ability to prove that basically statement p N is walking to be true girl and also greater 보다 or equal to 18. For this reason in part B remainder to prove the prize in component and making use of mathematical induction. So an initial of all, we have that 18. It's equal to six times three, so it follows that p 18 or basis situation is true. And now We'll mean that the explain PK is true for some K better than or same to 18. You want to present that PK to add one is true. Let's consider a pair cases, i suppose, the que can be created using only three cent stamps. This way Kate's going to be same to a Times three or part integer a, i beg your pardon in this situation is walking to need to be higher than or equal to six. Hey, then we have that K minus two. For this reason let's break this up in part cases, so K is equal to eight times 3 plus to win times, 11 for part injures A and B. And also now intend that the case because conform using three or an ext three cent stamps. So expect that a is going to be better than or same to three. Climate it adheres to that K to add one is same to a minus. Three times three. This gonna be three a minus nine. The should be 10. My mistake Plus and then we desire to add tens. This will be B plus warm times 10. Therefore we've shown that PK to add one is true in this case. Now the various other possibilities space knows the number three cent stamps is strictly much less than three. For this reason a has to be at many two typical that three cent stamps can contribute at many six. And since K is better than or same to 18 it complies with that need to kind with at the very least two cent stamps. Yeah, the B is going come be greater than or same to two. For this reason we can replace 2 10 cent stamps just 20 cents v 73 cent stamps, simply 21 cents to acquire k to add one sense. Therefore it adheres to that K add to one is going come be equal to Well, we're gonna be including seven three cent stamps. I would certainly be taken far to 10 cent stamps three times a add to Beecham's tennis K and also then seven times three minus two gyms. Tennis one therefore K add to one. So you've displayed that neither case statement PK add to one is true. And also so by the primary of mathematics induction, it complies with that PN is walk to be true roll and greater 보다 or same to 18 in component C. We're asked come prove the exact same statement part A the using solid induction this time. Therefore this time we have at ours base case p 18 p 19 p 20 year old True in component a and now we're going to suppose that ns 18 ns 19 up with p k room all true for part que greater than or same to 20. We want certain that PK plus one is true due to the fact that Kenya's an excellent equipped 20 it adheres to that PK minus two is true and so us can discover some A and B in the integers confident cheers in particular, such the K minus two equates to eight times 3 plus b times 10 for this reason it complies with that K to add Warren is going to be same to a plus one times 3 plus b times 10 so it complies with that statement. PK to add one is true and also therefore through the major of strong induction. It again complies with at the statement. PN is going to be true for all end greater than or same to 18