Answer:
Step-by-step explanation:
The Strong Induction Principle establishes that if a a subset S of the positive integers satisfies:
- S is a non-empty set.
- If m+1, m+2, ..., m+k ∈ S then m+k+1 ∈ S.
Then, we have that n ∈ S for all n ≥ k.
- <u>Base case</u>: Now, in our problem let S be the <em>set of positive numbers than can be written as a sum of distinct powers of 2</em>. Note that S is non-empty because, for example, 1, 2, 3 and 4 belongs to S:
This is the so called <em>base case</em>, and in the definition above we set k = 1. - <u>Inductive step</u>: Now suppose that 1, 2, 3, .., k ∈ S. This is the <em>inductive hypothesis.</em> We are going to show that k+1 ∈ S. By hypothesis, since k ∈ S, it can be written as a sum of distinct powers of two, namely,
where 
, i.e., every power of 2 occurs only once or not appear. Using the hint, we consider two cases:
- k+1 is odd: In this case, k must be even. Note that
. If not were the case, then 
and we can factor 2 in the representation of k: 
This will lead us to the contradiction that k is even. Then, adding 1 to k we obtain:
- k+1 is even: Then
is an integer and is smaller than k, which means by the inductive hypothesis that belongs to S, that is, 
where 
, for all 
. Therefore, multiplying both sides by 2, we obtain 
This is a sum of distinct powers of 2, which implies that k+1 ∈ S.
Then we can conclude that n ∈ S , for all n ≥ 1, that is, every positive integer n can be written as a sum of distinct powers of 2.
Answer:
557,143
Step-by-step explanation:
6.3% * x = 35,100
x = 35,100/0.063 = 557,143
Answer:
-1050
Step-by-step explanation:
I know the answer because descending means going down so 2100 is the big number and its negative to if you do -2100 divided by 2 then you will get -1050.
The least is 2/7 then 1/2 then 5/6 and the greatest is 4/2
Answer:
M = -2/5
Step-by-step explanation:
To find the slope of a line when you know two points, use the following formula.
M = y₂ - y₁ ÷ x₂ - x₁
M = 9 - 11 ÷ -12 - (-17)
M = -2 ÷ 5
M = -2/5
Hope that helps.
