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:
I highly recommend that it must be the second story
9514 1404 393
Answer:
- Purchase fewer than 15 jerseys from Print It; purchase more than 15 jerseys from Top Print; purchase exactly 15 jerseys from any supplier
- Value Printing could charge a $15 one-time charge and $20 per jersey
Step-by-step explanation:
1. For x jerseys, the cost from Print It is $21 each, so ...
p = 21x
The cost from Top Print is less per shirt, but has a set-up charge. Its cost can be modeled by ...
t = 45 +18x
These two costs will be the same when ...
p = t
21x = 45 +18x
3x = 45
x = 15
Purchasing 15 shirts from either supplier will cost $21·15 = $315. The lower per-jersey cost at Top Print will come into play if more than 15 shirts are ordered.
- Bill should order fewer than 15 shirts from Print It, and more than 15 shirts from Top Print.
- An order of exactly 15 shirts will cost the same from any supplier.
__
2. Value Printing wants their pricing to be somewhere between the prices of the other two shops. That means their per-jersey cost will be somewhere between $18 and $21, and their setup cost will be less than $45. It also means their cost for 15 jerseys must be exactly $315, as it is for both other shops.
If we choose a per-jersey cost of $20 for Value Printing, then the setup cost must be ($315 -$20×15) = $15 in order for the cost of 15 jerseys to match the cost at the other shops. With these choices, at Value Printing, ...
- pay a one-off setup cost of $15
- then each jersey will cost $20
Answer:
The answer is 207 ft^2
Step-by-step explanation:
Answer: they lost 16 games
<u>Step-by-step explanation:</u>
Wins: x + 17
Losses: x
Ties: 3
Total: 52
Wins + Losses + Ties = Total
x + 17 + x + 3 = 52
2x + 20 = 52
2x = 32
x = 16
