[Solved]Lemma 1 Total Circlyde Positive Circlyde Size Least 2 Must Pear Adjacent Elements 1 Mod N Q37045139

Lemma 1: If the the total of a circlyde is positive, andthe circlyde has size at least 2, then there must be a pear ofadjacent elements a(i) and a(i+1) mod n whose sum is positive suchthat ai is positive.  

PLEASE ANSWER ALL 4

1. Prove the Circlyde Pair Lemma using an proof technique youwish

Now, we will prove the Fundamental Theorem of Circlydesusing induction on the circlyde

2. State and Prove the Base Case

3. State the Inductive Hypothesis

4. Prove the Inductive Step

e define a circlyde a of size n to be an ordered list of numbers ao, at, 。。。 , an- arranged clockwise around a circle. Here are two examples of circlydes, named -1 -3 7 5 12 -5 4 2 -4 I-1,-5,4] size 3 [-3,7,12,-4, 2,5] size 6 elements a, and aj of a circlyde are adjacent iff ǐミj±1 (mod n). The total of a circlyde is the sum of all of the elements in the circlyde. So, the total of A is 19, and the total of B is -2. A cirelyde is called summy if there is some ar (k E N,0 k< n) such that all of the partial sums moving clockwise around the circle that start from ak are positive. More formally: i-k (Here, s and k are somewhere between 0 and n – 1, inclusive. Also, note that the mod operator in the sum appears in the subscript – this is to allow the sum to continue wrap- ping around the circle.) For example, circlyde A above is summy, because if we start from a2 = 12, we see that 12, 12- 4, 12-4 + 2, 12-4+2+ 5, 12-4 2 + 5 -3, and 12-4+ 2 +5-3 + 7 are all positive. Circlyde B above is not summy, because the total of B is negative (so no matter which index you choose for k, the sum where s = n-1 will be negative). On the next two pages, we will use induction to prove the following theorem: Theorem 1 (Fundamental Theorem of Circlydes) If the total of a circlyde is pos- itive, then the circlyde is summy. Show transcribed image text e define a circlyde a of size n to be an ordered list of numbers ao, at, 。。。 , an- arranged clockwise around a circle. Here are two examples of circlydes, named -1 -3 7 5 12 -5 4 2 -4 I-1,-5,4] size 3 [-3,7,12,-4, 2,5] size 6 elements a, and aj of a circlyde are adjacent iff ǐミj±1 (mod n). The total of a circlyde is the sum of all of the elements in the circlyde. So, the total of A is 19, and the total of B is -2. A cirelyde is called summy if there is some ar (k E N,0 k

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Answer to Lemma 1: If the the total of a circlyde is positive, and the circlyde has size at least 2, then there must be a pear of… . . .