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numerator of nth term   (2n)!

 numerator of n+1th term (2*(n+1))!  =  (2*n + 2*1) ! = (2n + 2)! = (2n+2)*(2n+1)*(2n)!

it's wrong!! :(((( 

Need to find  'k'  and  'a'

at t= 0  infected = 5000

5000 = 20000/(1+ae^-k(0))     yields a = 3

Now find 'k'

At t= 4    infected = 8923

8923 = 20000/(1+3e^-k(4))     Yields  k = .2205934

At t= 8

f(t) =   20000/(1+3e^-.2205934(8))      = 13213 people

When are 12192 infected?

12192 = 20000/(1+3e^-.2205934t)       Solve for 't' weeks.....I'll let you finish     Hint :   should be less than the 8 weeks calc'd above !!

percy jackson

the answer is wrong

Because \(\angle Q = \angle R\), they must be acute angles(because if they were the right angle, then two right angles would already be 180 degrees, making the triangle degenerate). Name the angle measurement of one of these 2 angles x. We can write the expression:

90 + x + x = 180(the internal angle sum of a triangle).

90 + 2x = 180

2x = 180

x = 45

That means that we have a 45-45-90 triangle at hand, which happens to be a special right triangle. The ratio of a 45-45-90 triangle is:

\(1,1,\sqrt2\) where the legs are 1, and the hypotenuse is \(\sqrt2\)(Remember this! It will make math with certain right triangles a lot easier!)

We know that angle P must be the right angle, because angle Q and R are both 45 degrees.

We then have that the hypotenuse PR is equal to \(6\sqrt2\)

Because the ratios of a 45-45-90 triangle are 1,1,\(\sqrt2\), we know that right triangle PQR is scaled by a factor of 6 compared to a 45-45-90 of leg length 1.

That means the leg lengths of our triangle are:

6 * 1 = 6

Now that we have the length of the legs as 6, we just need to find the area.

The area is equivalent to the (height * base) /2. Since we have both legs, we can just set one of them as the height, and the other the base(since they share a 90 degree angle). We then have our area as:

(6*6)/2 = 18

Name your desired number as x. If 20% of x is equal to 81, we can write:

\(0.20x = 81\)(do you see how I got here?)

The way you reach this expression is because 20% is equivalent to 20 * 1/100, which is just 20/100, which is equivalent to 0.20

Then, we realize that if we multiply this by 5, we just get:

\(x= 405\)

That gives us our desired answer.

The x intercept is just the point on the line at which the line intersects the x axis. In other words, the y value is equal to 0(can you see why?). When the Y value is equal to 0, that means that the line has to intersect the x axis because the x axis is at a y value of 0. Similarly for the y intercept, it's defined as the point that intersects the y axis on a given line. That means that the x value is 0, because the y axis is located at x = 0. That means we just have to look for the points that intersect the x and y axes respectively. We see that there is one point on this graph that intersects both, which is directly at the origin. This gives us an x intercept of 0, and a y intercept of 0, because the coordinates of the origin are (0,0)

no problem

thank U

Notice that the legs of the right triangle are in a ratio of 3 : 4 . If you remember anything about right triangles, you'll know the "3 - 4 -5" right triangle, which is a right triangle of side lengths 3,4, and 5(is there another way to phrase this ? lol). We can see that this right triangle of legs 20 and 15 inches is a "scaled up" version of the 3 4 5 right triangle, and is in fact, similar to the 3-4-5 triangle(by a ratio of 5). As such, we know that the hypotenuse of this triangle must be 5 * 5 = 25 inches long. Alternatively, we can just use pythagorean theorem to find the length of the hypotenuse of our triangle. Using pythagorean theorem, we have where h is the hypotenuse:

\(h^2 = 15^2 + 20^2 = 225 + 400 = 625\)

\(h = \sqrt{625} = 25\)

That means that the perimeter of said right triangle is:

\(15+20+25 = 60\)

Because the square and right triangle both have equal perimeters, we can name the side length of the square x

We can then write the equation:

\(4x = 60\), because the square has 4 sides of length x, which gives us its perimeter of x + x + x + x = 4x

Dividing by 4 on both sides, we get: 

\(x = 15\)

The area of a square is equal to 

x² given a side length x, so we then have:

\(15^2 = 225\) in² as the area of our square

Sorry I don't have an accompanying diagram(I'm not too tech savvy with programs that others are more familiar with like desmos, etc.), but I'll try my best to explain my solution.

We draw a large circle, and within it, another smaller circle that shares the same center(that's the definition of concentric more or less). Then, we draw a chord(a line segment with both endpoints on the circle) across the large circle that touches the smaller circle at exactly one spot. After we've done this, we can then draw a line from the center of the smaller circle to the point of tangency, creating a 90 degree angle. By definition, if a line is tangent to a circle, it forms a 90 degree angle with the radius drawn to that point.

Here's a proof of the following done by Euclid:

https://www.quora.com/Why-does-the-tangent-of-a-circle-make-an-angle-of-90%C2%B0-with-the-radius-drawn-from-the-point-of-contact

Next, we notice that we can draw in two radii of the larger circle, both of which intersect an endpoint of the chord. We then form two right triangles that have the same hypotenuse and same leg(which is the radius of the smaller circle). By pythagorean theorem, it stands to reason that if the hypotenuse and leg of a right triangle are congruent to another hypotenuse leg pair, then both right triangles are congruent. For this reason, we know the legs of the two right triangles are equal to each other, meaning that the point of tangency is indeed the midpoint of the chord drawn.

Getting an ace would have a probability of 4/16, or 1/4.

When someone draws one, one is taken away.

The equation would be: 1/4 times 1/5 times 1/7 times 1/13.

Hope this helped!

Anytime :D

the end of the question is a bit different :)

I still feel very confused.. I'm sorry. I got lost at the 1+n^th term. Why would you add the 2 on both sides of the fraction? 

After that, I get the work. I tried solving for myself and I got the same answer. So that's the only question I have. 

Thank you for helping, by the way!

Thank you as well, Alan!

Thank you so much for your time and effort :D!

forty nine   is   7²

Good work jfan17. However, you neglected something very important. The question is asking "Who gets more INTEREST after 3 years". So, you will have to subtract the principal from the calculated amounts in each category to see who earned the most INTEREST. I believe Sue has more interest in part "a" and Bill has more interest in part "b".