Please do number 3 with full work and explanation. Thanks
This workshop deals with an addition formula in section 5.1. This formula is related to an oft told story about the great mathematician Gauss. Here is one version of the story When Gauss was ten years old, he attended a certain math class with other kids. One fine day, the math teacher discovered that he was running out of time to grade the latest b of math homeworks. This is why he decided to finish the grading during his math lecture period. To keep the students busy while he graded, he gave them the following task at the start of the period: Use your slate (the s in class) to add up the numbers 1, 2, 3,... , 100 Seconds after the teacher gave this in-class assignment, little Gauss pointe and yelled "Here is the answer." Then Gauss did nothing for an hour, while the oth atch mall personal blackboard that every student had d to his slat At the end of the period. showed the correct answer 5050 on his slate. Only Gauss got the correct answer Then Gauss told everybody that the surn 1 + 2 + 3- + 100 is (1 +100)+(2+99) +(3+98)+(50+51) 101 +101+101+ +101 101 50 5 Problem 1. Explain carefully what Gauss had done. Include the reason why it is impor tant that 100 happens to be an even numbet Problem 2. Let N be an even positive integer. Use the method of Problem the formula 1 +2+3+... the be an even number Problem 3 The formula is still valid when N is an odd integer. Prove t variation on the argument given by Gauss The story continues: Gauss then gave the class to sum an arithmeric series (a+1.b) + (a+2.) + (a +3.b) ++ and b are constants N . Include the reason why it is important that a lecture on how his method could be used a+N .), where a