▀▄▀▄▀▄ 2nd ѕeмeѕтer ѕυммary ▄▀▄▀▄▀

Greetings, guest of Mathland!

This will be the last post of the year!~ 

Thank you so much for accompanying me throughout my journey.

Starting off in the beginning of the year, we began with lessons on pre-calculus. As the year progressed, we peeked into the world of Calculus through the lessons of parametric equations, limits, and etc.

Overall, the year was a lovely and I was able to learn about numerous new concepts. Although, sometimes a few were confusing, but later on through practice, I was able to master them! 

Something I enjoyed the most in math this year would be storytime - as it was very relaxing during stressful times (AP Week game strong).

There is nothing I didn't like about the class....other than webassign which was a bit stressful occasionally. (o~ o)

But overall, I loved the class!

Thank you again!

(¯`·.¸¸.-> °º тяιg яєνιєω ωєєк º° <-.¸¸.·´¯)

Hello, guests of Mathland!

This week, we had trig review week which went back to our trig identity proofing. 

To start off, we began with simplifying the trig expression in order to make the problem easier to do.  After that, we verified (or proved) each identity - meaning we proved that the left side equaled the right side.

To do such a thing, we used a trigonometric identity sheet that had all the identities on it:

Learning these identities again were a bit difficult, as these were taught near the beginning of last semester. However after a bit more practice, it was easier to do!

Below is an example of simplifying a trig function:

Below is an example of proving a trig identity:

That is all for today, good bye! See you next time!~

••.•´¯`•.•• ŖËPËÄȚÏŅĠ ĐËĊÏMÄĻŚ ••.•´¯`•.••

Salutations, guests of Mathland!

Today we will be learning a fairly simple topic - repeating decimals!

Examples of such decimals are:

.23232323....

.7777...

.417417417....

It is stated that every repeating decimal is the sum of an infinite geometric series.

To solve for the repeating decimal, you would first write the decimal as a geometric sum. Then you would find the sum.

The equation used in this case is a1/(1-r).

To find r, it would be the common difference or ratio between each number in the repeating decimal.

Below is an example:

That is all! See you next time!

ıllıllı ραяαмεтяιc εqυαтισηs ıllıllı

Greetings, guests of Mathland!

Today, we will be reviewing parametric equations!

These equations give you the starting point and direction a particular line will flow. Starting off, the first step to solving the parametric equation is by sketching a graph. To sketch the graph, you would need to make a table that used the variables of:

 t,x, and y (which are included in the given equations).

For example:

After that, you would usually solve the equation to eliminate the parameter. There are two different ways to approach, by using either elimination or substitution, or the trig identities.

Four important trig identities that should be memorized are:

Sometimes, instead of substituting regular whole numbers for t, the equation will ask for degrees, in which you will need to refer to the unit circle.

Below is an example of solving a parametric equation:

That is all. Ta ta!~

—(••÷[ ⓟⓐⓡⓣⓘⓐⓛ ⓕⓡⓐⓒⓣⓘⓞⓝⓢ ]÷••—

Welcome, guests of Mathland!

Today we will be learning about partial fractions. 

In starting off, there are a few set up steps that must be written out.

Here are the steps:

1) Divide if improper

2)Factor denominator

3) If linear (no exponent), it is A,B, C...

4) If quadratic then Ax+B, Bx+C, etc...

To solve for the equation:

1) Multiply by the LCD (Least Common Denominator

2) Group terms by powers of x

3) Solve the system of equations

Further more, there are 4 cases that encompass partial fractions. The first is linear, the second is quadratic, the third is squared on the outside, and the last is long division.

Below are examples of the different cases:

Case 1:

Case 2:

Case 3:

That is all. Farewell!~

▁ ▂ ▄ ▅ ▆ ▇ █ тower oғ нanoι reғlecтιon █ ▇ ▆ ▅ ▄ ▂ ▁

Starting off, within the concept of mathematical induction, there are two major steps. The first is "Show true for n=1" and the second is "Assume the statement is true for n=k, and prove true for k+1." 

Using such steps in an example...

If the equation or problem given is 1 + 2 + 3 +4... + (2n-1) = n^2 , one would first begin by proving that n=1. To do this, you would plug in a "1" for anywhere you see a "n" within the equation. Therefore it would look 2(1) - 1, which is equal to 1 - and then (1)^2, which is also equal to one. By doing that, we have proved that both the left hand side and right hand side are equal to 1, showing true that n=1.

After that, we would continue on to step 2, which is "assume the statement is true for n=k, and prove true for k+1." This means that first, we will plug in a "k" for anywhere we see a "n" in the problem. Upon doing so, the resulting equation would be:

 1 + 2 + 3 +4... + (2k-1) = k^2  (This is known as the recursive formula)

After you find the equation above, you then plus in a "k+1" for anywhere a k is present. Therefore, the next resulting equation would be:

 1 + 2 + 3 +4... + [2(k+1)-1] = (k+1)^2

You then simplify the equation and plug in the recursive formula (mentioned above). This indicates that the equation will become:

k^2 + 2k +2-1 = (k+1)^2

and then simplifying it down

k^2 +2k + 1 = (k+1)^2

which is

(k+1)^2 = (k+1)^2

Thus we have shown that k+1 is true. It follows from Mathematical Induction that the statement is true for every positive integer.

-----

Moving along, in class, we did a project called "The Tower of Hanoi." This particular project was to find the correlation between the number of rings and the amount of times it was moved. Starting with 2, then 3, then 4, and then maxing at 7 rings, with three poles; my group discovered that the equation for the correlation was:

T1 = 2^n - 1

and the recursive of it was:

Tn = 2Tn +1

An observation i concluded while experimenting with the discs is that if you move the lowest number to the second pole, and then move the second lowest number to the third pole, you can then put the one on the second pole on the third pole - therefore, giving room for the third to be moved to the second pole. By using such a strategy, it is easier to move all the discs from the first pole to the third pole.

—(••÷[ ŚËǬŮËŅĊËŚ ÄŅĐ ŚËŖÏËŚ ]÷••—

[I will be simplifying the blogs from today on so that I may save time to study for upcoming AP's ]

Greetings, dear guests of Mathland!~

Today I will be expanding on sequences and series. For sequences and series, there are two types:
-arithemetic
and
-geometric

For the arithemetic sequence, the formula used is an=a1 +(n-1)d. "D" is the ratio between each number - otherwise known as the common difference.
For the geometric sequence ,the formula used is an=r^(n-1)
In addition to using these equations, one is frequently asked to find the sum of the series, meaning what is the sum of the pattern added up to a specific number. To find the number, summation notation is used.
The arithmetic formula for this is Sn = n[ (a1+an)/2 ]
The geometric formula for this is Sn = [a1(1-r^n)]/1-r
Here is an example:

That is all for today! Good bye!~