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.
The area of a rectangle is found using the formula A=LW. You need to express the length and width of this table using one variable. Since the length is described in terms of the width, let's call the width w. The length is 3m more than the width, so we can call the length w+3. Put these expressions of the length and width into the formula:
<span>A=LW, A=88m</span>²<span>, L= (w+3), w=w substitute: </span>
<span>88=w(w+3) (ok I did WL instead of LW but it's the same thing) Distribute the w: </span>
<span>88 = w</span>²<span>+3w subtract 88 from each side </span>
<span>0 = w</span>²<span>+3w-88 factor. w^2 can only be factored by w*w so set that up: </span>
<span>(w )(w ). Now look at the last term, it is -88. You need to find two factors of -88, and since factors multiply to get the number, you must have one positive and one negative factor to get -88, so go ahead and put them into your parentheses: </span>
<span>(w+ )(w- ). Now find factors of -88 that when added together give you 3, the coefficient of the middle term. -8 and 11 add up to 3, so put those into your factors: </span>
<span>(w+11)(w-8) and remember that this equals 0: </span>
<span>(w+11)(w-8)=0 Now when you multiply two numbers and get a product of 0, at least one of those numbers must equal 0 since any number times 0 equals 0. Either one can be 0, so set each equal to 0: </span>
<span>w+11=0 subtract 11 from each side </span>
<span>w= -11 </span>
<span>w-8=0 add 8 to each side </span>
<span>w=8 </span>
<span>Now you're asked to give the length and width of the table top. The table can't have a dimension that is negative, so we can discard the solution w= -11. We find that the width of the table is 8 feet. To find the length, go back to how we expressed the length, 3 more than the width: </span>
<span>L=3+w, w=8 substitute </span>
<span>L=3+8 add </span>
<span>L=11. </span>
<span>So the length of your table top is 11m and the width is 8m.</span>
Answer:
YesUs
Step-by-step explanation:
the vertical LIne test. The lines intercept, so it is not a function.
67+79+80+190= 416
Each number has been rounded
