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When Hubble identified some 'Cepheid variable' stars in the Andromeda Galaxy, it allowed him to show the distances to these stars were greater than Shapley's proposed size of our galaxy (Curtis's estimated size was smaller then Shapley's) allowing a conclusion that it was another galaxy much like our own. This proved Curtis right that there were many spiral galaxies just like our own rather then just one as Shapley hypothesised. Shapley however was later proved to be closer to the correct value for the size of our galaxy and our sun's location in our galaxy. 

Cepheid variable stars are used as cosmic benchmarks of galactic and extragalactic distances.

Physics

What event settled the Shapley-Curtis debate about spiral nebulae?

This is a hard nth term question!
We will solve with method of differences.

4,7,10,16,28
First difference (relates to a linear sequence if constant n): 3,3,6,12
Second difference (relates to a quadratic if constant n^2): 0,3,6
Third difference (relates to a cubic if constant n^3): 3,3

The third difference is constant (though we only have two terms of it, if this wasnt constant, we couldn't easily solve nth term for this part of the sequence).

Since third difference is constant, we know it is cubic of form an^3+bn^2+cn+d = nth term where a,b,c,d are constants.

We can also do this with 5 simultaneous equations from here on out.
first term when n = 1:                a + b + c + d = 4 [1]
second term when n = 2:      8a + 4b + 2c + d = 7 [2]
third term n = 3:                       27a + 9b + 3c + d = 10  [3]
fourth term n = 4:                   64a + 16b + 4c + d = 16 [4]
fifth term n = 5:                      125a + 25b + 5c + d = 28 [5]

I have labelled all equations with square brackets (not part of the actual equation).

Using [2] - [1]:   7a + 3b + c = 3    [2-1]
Using [3] - [2]:  19a + 5b + c = 3   [3-2]

Using [2-1] - [3-2]:   12a + 2b = 0  [A] Relabelled as letters

Using [4] - [3]: 37a + 7b + c = 6  [4-3]
Using [5] - [4]: 61a + 9b + c = 12  [5-4]

Using [5-4] - [4-3]:  24a + 2b = 6  [B] Relabelled as letters

[B] - [A]: 12a = 6 therefore a = 1/2
If a = 1/2 using [B]: 12+2b=6 therefore b = -6/2 = -3
Then using [2-1]:  7a + 3b + c = 3 
which substituting a and b in goes to 7/2 + (3x-3) + c = 3
therefore c = 3 + 9 - 7/2 when you rearrange and expand
c = 17/2

Finally lets use [1] to find d. 
a + b + c + d = 4
1/2 - 3 + 17/2 + d = 4
d = 4 - 18/2 + 3
d = -2

You should be able to get to this result using different combinations of these equations. Just make sure you use as many as you can. If you try and cancel to many terms using the same information (e.g same equations). You'll likely just end up with something like 0 = 0.

Therefore nth term is combination of a,b,c,d as we defined earlier.
an^3 + bn^2 + cn + d = 1/2(n^3) - 3(n^2) + (17/2)n  - 2

Sorry for such a long solution.
I think there are other ways to get to this solution. It is quite a tricky one and easy to make mistakes in the workings. You can test this solution and I think it works

Maths

What Is nth term of the sequence 4,7,10,16,28 ?

I have written a full algebraic solution to see this, but there is also a quick way of thinking this through at the bottom.

Let's take the area of rectangle A and let us call it - a. 
Similarly we will call area of rectangle B - b and the same for rectangle C - c.

In the question we know area b is 10% bigger then area a. So we can write.

<Eq 1>
b = 1.1a (or we could write 100a = 110b if we want to work with the percentages - these are equivalent)

<Eq 2>
Similarly we know:
c = 1.1b (or 100c = 110b)

Let's rearrange this so we can substitute into <Eq 1>
For this we want b by itself on a side. Therefore let's divide both sides by 1.1.

c / 1.1 = b

Now let's substitute b in to <Eq 1> to allow us to relate a and c.
c / 1.1 = 1.1a

Multiply both sides by 1.1
c = (1.1 x 1.1)a
c = 1.21a

To use percentages let's multiply 1.21
So 100% of c - the full area of c is equal to 121% of shape a. The area has increased by 21%. 

Instead of writing the full algebraic solution, you could think of it as shape a increasing by 10% and then this increasing by 10%. In decimal this is a 1.1 squared increase (1.21). 





There are 3 rectangles A B and C. The area of rectangle B is 10% greater then the area of rectangle A. The area of rectangle C is 10% greater then the area of rectangle B. By what percentage is the area of rectangle C greater then rectangle A?

This is an example of a two circle Venn diagram. Not easiest to explain here but I'll try my best. Since you have two categories, it will have two large circles. Since these are not mutually exclusive (they could use both), the circles will overlap in the middle. You know 150 use neither, so you can write this outside the circles immediately. Now you know you had 605 employees but you wrote 150 outside the circles therefore you now only have 605-150 that can be inside = 455. You had 285 using the cafe and 220 using the gym. Now some of these could be in the overlap. 

Let's set people only using the cafe = c, people using only the gym = g, and people who use both = b.

We know 285 = c + b since 285 people use the cafe (includes the people who only use cafe and people who use both). 
We know 220 = g + b for the same reasoning as above.
Let's add these together (as for simultaneous equations) [let me know if you don't know how to use these].

505 = c + 2b + g

We know 455 = c + g + b (total inside the circles of Venn diagram)
Therefore, by subtracting these two we get 50 = b. This means our overlap contains 50 people.

Using our previously defined equations:
285 = c + b
220 = g + b
we can now work out c and g since we know b.

c = 235
g = 170

Therefore, our Venn diagram contains 235 people in the circle labelled Cafe, 170 people in the circle labelled Gym, 50 people in the overlap and 150 outside both circles. Let's add these up to check our answer.

235+170+50+150=605 => Everyone is accounted for. Our answer should be correct. 

Hope this helps!

How can i answer this question "In an office complex 220 of 605 employees use gym, 285 use cafe and 150 use neither. Represent this in a venn diagram." Also what is the probability of someone randomly selected uses both facilities?

In terms of the hydrosphere, farming can cause nutrient deposition. For example - fertilizer runoff/leaching. Often fertilizers are applied in excess. This nutrient deposition into waters can cause processes leading to eutrophication (algal blooms) and hypoxia (lack of oxygen). Other sources from farming of excess nutrients can be overuse of manure, aquaculture (fish farming), etc.





Biology

How does farming affect the biosphere atmosphere and hydrosphere?

Electrolysis is the splitting of ionic substances using electric currents. The electric current causes the movement of positive and negative ions. Positive ions will travel to the negative electrode where they are reduced (gain electrons) and negative ions move to the positive electrode where they are oxidised (lose electrons).

Science

Can someone help explain the process of electrolysis?

In simplest terms it is that everything can act as a wave and also as a particle. Everything will have a wave function. 

What is the wave particle duality in quantum physics?

1/2 x base x perpendicular height
or
1/2 x a x b x sinC where C is the angle between two sides length a and b. 

These two formulas are actually the same since b x sinC would give you the perpendicular height.

What is the area in a triangle

You can formulate this with algebra.

If the original numerator is x, then the denominator will be 3x-1.
Next add two to both these, leaving us with x+2 as numerator and 3x+1 as denominator. Now rewrite these as a fraction and equate to 1/2. 

Now solve
(x+2)/(3x+1)=1/2
2x+4 = 3x+1
x=3

You can now resubstitute into original equations for numerator and denominator, giving a fraction of 3/8.

The denominator of a fraction is 1 less than 3 times the numerator. If the numerator and the denominator are both increased by 2, the fraction becomes 1 over 2. how can I find the original fraction?

Alexander Stronach

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