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Proofs
Proof means sufficient evidence to establish the truth of something. It is obtained from deductive reasoning, rather than from empirical arguments. Proof must show that a statement is true in all cases, without a single exception.
1,294 Questions
No, it comes out with a decimal, 814 divided by 4 equals 203.5
How do the squares of sums and differences relate?
(a + b) = a^2 + 2ab + b^2
(a - b)^2 = a^2 - 2ab + b^2 or you can work like this:
[a + (-b)]^2 = a^2 + 2a(-b) + (-b)^2
(a - b)^2 = a^2 - 2ab + b^2
How do you calculate the nth term in the Fibonacci sequence?
For low terms, it is probably most economical to to simply list out the sequence until you arrive at the nth term, but for large terms, there actually is a closed-form equation.
Let (phi) = (1 + sqrt(5))/2. As an aside, this is the golden ratio. Then the nth term of the Fibonacci sequence is given by
F(n) = [(phi)^n - (1 - (phi))^n]/sqrt(5),
where we are letting F(0) = 0, F(1) = 1, F(2) = 1, etc.
The derivation of this formula involves linear algebra, and, in short, the key is in setting this up as a matrix equation and diagonalizing the matrix. While I shall not run through the process here, I will give you the matrix and vector to start with:
Let A be the 2x2 matrix
[0 1]
[1 1]
and x(n) be the column vector in R^2
[F(n)]
[F(n+1)]
In particular, x(0) = (F(0),F(1)) = (0,1).
notice that [A][x(n)] =
[F(n+1)]
[F(n)+F(n+1)]
which is x(n+1). Thus, it follows that x(n) = [A]^n[x(0)] (you can prove this rigorously through induction). Now, F(n) is the first term of the vector x(n), so what one needs to do is diagonalize A, raise A to the appropriate power, multiply it with x(0), and finally take the upper term. The result is what I presented at the outset.
Note that with this formula you can show that lim(n-->infinity)[F(n+1)/F(n)] approaches phi.
Interval notation uses the symbols [ and ( to indicate closed an open intervals.
The symbols can be mixed so that an interval can be open on one side and close on the other.
Given two real numbers, a, b
we can have
(a,b) which is the interval notation for all numbers between a and b not including either one.
[a,b) all numbers between a and b including a, but not b.
(a,b] all numbers between a and b including b, but not a.
[a,b] all number between a and b including a and b.
In parallelogram the diagonals have the same midpoint?
Yes the diagonals of a parallelogram have the same midpoint since they bisect each other.
true
Positive numbers have two square roots a principal square root and its?
And its negative counterpart.
The positive integer factors of 210 are 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, and 210
The factor pairs of 210 are (210,1), (105,2), (70,3), (42,5), (35,6), (30,7), (21,10), and (15,14)
Why is pride a feeling and not a state or quality?
I'd say that pride is more akin to a feeling as it is based on one's individual perception of self, and can differ from how the world views oneself. Because everybody has a differening sense of pride, I'd definitely argue that it is a feeling.
Ceorgivs v graomnrexfiddefindimpplease tell if its worth keeping?
More information is needed. Your coin is almost certainly British, but you also need to supply its denomination and condition. Please post a new question so that it will be possible to ID your coin.
The motto is several words, not one, and variations of it are found on all British coins. It's heavily abbreviated Latin for "George the 5th, by the Grace of God King of All Britain, Defender of the Faith, and Emperor of India"
Select the postulate or theorem that could be used to prove QRT STR?
Well, this will depend on the length of the sides of the triangle for what postulate or theorem you will be using.
Do you mean circle?
The area of a circle is pi times the radius2.
How do you solve lntan(x)lnsin(x)-lncos(x)?
How do you solve ln|tan(x)|=ln|sin(x)|-ln|cos(x)|? Well you start by........
How long does it take to count till ten in ten seconds?
It takes exactly ten seconds to count to ten in ten seconds.
What does this phrase more opinions than you can shake a stick at mean?
It means that you have so many choices to go by.
Prove that A contains N elements and the different subsets of A is equal to 2?
Assuming the question is: Prove that a set A which contains n elements has 2n different subsets.
Proof by induction on n:
Base case (n = 0): If A contains no elements then the only subset of A is the empty set. So A has 1 = 20 different subsets.
Induction step (n > 0): We assume the induction hypothesis for all n smaller than some arbitrary number k (k > 0) and show that if the claim holds for sets containing k - 1 elements, then the claim also holds for a set containing k elements.
Given a set A which contains k elements, let A = A' u {.} (where u denotes set union, and {.} is some arbitrary subset of A containing a single element no in A'). Then A' has k - 1 elements and it follows by the induction hypothesis that (1) A' has 2k-1 different subsets (which also are subsets of A). (2) For each of these subsets we can create a new set which is a subset of A, but not of A', by adding . to it, that is we obtain an additional 2k-1 subsets of A. (*)
So by assuming the induction hypothesis (for all n < k) we have shown that a set A containing kelements has 2k-1 + 2k-1 = 2k different subsets. QED.
(*): We see that the sets are clearly subsets of A, but have we covered all subsets of A? Yes. Assume we haven't and there is some subset S of A not covered by this method: if S contains ., then S \ {.} is a subset of A' and has been included in step (2); otherwise if . is not in S, then S is a subset of A' and has been included in step (1). So assuming there is a subset of A which is not described by this process leads to a contradiction.
What is five plus five minus five?
A.
5+5-5=
10-5=
5
..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... .....
B.
5+5-5=
5+0=
5
..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... .....
C.
5+5-5=
-5+5+5=
-5+10=
5
congruent halves are just a posh way of saying 'split in two'. but these two shapes must be identical in shape, size, angles, sides etc... they must be exactly the same.
Prove that angle subtended by chord at a point?
The question cannot be answered because it is not clear what is to be proved.
Theorem
What is the proof about multiplying twin primes and adding 1?
The question depends on what it is that you want to prove!
