Westonci.ca is the Q&A platform that connects you with experts who provide accurate and detailed answers. Get immediate and reliable answers to your questions from a community of experienced professionals on our platform. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
Answer:
It’s true
Step-by-step explanation:
Proof by induction
Let the four chosen numbers be x_1, … x_4 and, without loss of generality, let x_1 < x_2, … < x_4.
For x_1 = 1, the smallest possible choice, let x_2 = x_1 + 20 = 21 be the smallest possible choice that has a difference larger than 19. Likewise, x_3 = x_2 + 20 = x_1 + 40 = 41. Then x_4 has to be at least x_3 + 20 = x_1 + 60 = 61 which is out of range.
For the induction, suppose that x_4(n) = x_1(n) + 60 > 60 and therefore x_4(n) can’t be chosen for a given x_1(n). Then incrementing x_1(n+1) = x_1(n) + 1 would require x_4(n+1) to be at least x_1(n) + 1 + 60 = x_4(n) + 1 > 61, so this is out of range as well.
For another induction, suppose that x_4(n) = x_2(n) + 40 > 60 and therefore x_4(n) can’t be chosen for a given x_2(n). Then incrementing x_2(n+1) = x_2(n) + 1 would require x_4(n+1) to be at least x_2(n+1) + 40 = x_4(n) + 1 > 61, so this is out of range as well.
Likewise, for the induction, suppose that x_4(n) = x_3(n) + 20 > 60 and therefore x_4(n) can’t be chosen for a given x_3(n). Then incrementing x_3(n+1) = x_3(n) + 1 would require x_4(n+1) to be at least x_3(n+1) + 20 = x_4(n) + 1 > 61, so this is out of range as well.
Therefore all choices of x_1, x_2, x_3 with differences greater than 19 leaves no choice of x_4 within range with x_4 > x_3 + 19.
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.
