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# Week 3 in Pre-Calc 11

Something I’ve learned this week is about absolute values.

Absolute value doesn’t really have a definition. Technically, absolute value is how far away a number from zero is.

The symbol for absolute value is: ∣

And the value should be inside of those straight lines. E.g. x

So, this is how it works. First, let’s look at this number line… As we can see, “3” is 3 lines away from zero and “-3” is also 3 lines away from zero.

So| 3 | (read as: the absolute value of 3) is 3 and | -3 | is also 3.

Why? Because recalling the technical definition of absolute value, it says “how far” a number is from zero and it’s not asking in which direction it’s from zero. In other words, the absolute value of a number is always positive!

The technical definition should also be like this: Meaning, the square root and the power of 2 (or square) will cancel each other out.

NOTE: Absolute value doesn’t work like parentheses. | | should not be mistaken as ( ).

For example, -(-2) is NOT -|-2| because:

-(-2) = +2

-|-2| = -(2) = -2

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# Week 2 in PreCalc 11

One of the things I learned in Pre-Calc 11 on the second week is about how to add an arithmetic pattern or sequence. I find it awesome because you can know the sum of a whole sequence without adding them all together.

An arithmetic sequence is a sequence that is increased or decreased by the same amount of value.

For example, let’s say everything adds up by two(2): 3 → 5 → 7 → 9 → 11 → …

Now, that’s a sequence. What we increase or decrease by is called the common difference (d).

But how do we get the sum of these terms without adding them up together?

Before we get to that, when we add the terms of an arithmetic sequence together is called arithmetic series.

The difference when you write a sequence and a series is that:

sequence is written like this: 3, 5, 7, 9, 11,…

series is written like this: 3 + 5 + 7 + 9 + 11 + …

So how do we add them altogether? Here’s the formula: $S_n = \frac {n}{2} (a + t_n)$

n is how many your terms in a sequence is. $t_n$ is your last term.

Let’s use my previous example before as the series that we’re gonna use and let’s put the 50th term as our last term.

so it means that … 3 + 5 + 7 + 9 + 11 + …. + 101 That means that our first term is 3, our last term is 101, and the number of our terms is 50.

Plug them all in to the formula and remember: PEMDAS. And we got it! The sum of all terms is 2,600.

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# My Arithmetic Sequence

Sequence: $4,9,14,19,24$

Formulas: $t_n = t_1 + (n-1)d$ $S_n = \frac{n}{2} (t_1 + t_n)$ $S_n = \frac{n}{2} (2t_1 + (n-1)d)$

Skill check:

• Find $t_{50}$ and give the general equation for $t_n$

a = 4                 d = 5

general equation: $t_n = 4 + (n-1) 5$ $t_n = 4 + 5n - 5$ $t_n = 9 + 5n - 5$ $t_{50}$ $t_{50} = 4 + (49) 5$ $t_{50} = 4 + 245$ $t_{50} = 249$

• Determine the sum of the first 50 terms.

a = 4       d = 5 $t_n = 249$ $S_{50} = \frac{50}{2} (4 + 249)$ $S_{50} = 25 (253)$ $S_{50} = 6325$