I ended up getting more done this week than I thought I would. I got through the unit Matrices on Wednesday, which I was surprised but happy about to have gotten through so early in the week, and then managed to pass the unit test of the following unit, Series, on my first attempt on Sunday morning. I wouldn’t say I understand either matrices or series through-and-through, BUT I do feel like I have a fairly firm grasp on both subjects, at least up to what I’ve learned so far. On another note, I’m fairly certain I have covid so this will likely be a shorter post. On Friday I noticed I had a sore throat and started feeling like I had low energy/weak/achy and still feel that way now. I’m not too worried about it if I do have covid but still do feel like shit so I’m not going to bother trying to make this post very long.

I kept working on inverse matrices this week and realized part of the equation had a name to it that I didn’t know about:

I knew about the ‘determinant’, which in a 2×2 matrix is (a * d) – (b * c), but didn’t know the name for the second part of the inverse matrix equation which, as you can see in the photo, is called the ‘adjunct’. This is where you swap a and d in the matrix and switch the (+/-) sign on elements b and c.

As I mentioned in the intro, I finished the Matrices unit test on Wednesday morning which had 14 questions and took me ~40 mins. I knew what formulas needed to be used in each question and how to use them but, even still, I didn’t feel 100% confident as I went through them. Because many of the formulas required multiple steps, even though they were all relatively simple calculations, I always felt like I might have made a simple mistake somewhere along the way. I think I need more practice with these types of questions to get more confidence going through them. That being said, I was happy with how the test went and felt good about my understanding of matrices and good about moving on to the next unit, Series.

Looking back in my blog post notes, I just realized I was first introduced to series 15 months ago back in Week 11 which is crazy to think about. The first thing this unit Series went through was the difference between a sequence of numbers and a series of numbers:

There are two types of sequences/series that I’ve learned about so far which are ‘geometric’, which are exponential, and ‘arithmetic’, which are linear. I first went through geometric series and their notation and formulas. Before learning about the formula to find the sum of a geometric series, it’s useful to know and understand the formula to find the value of the n^th element in a geometric series (i.e. {2 + 6 + 18 + … + ‘n’}):

From the photo, the first term of the sequence is denoted with the letter a. The letter r is used to denote the ‘common ratio’, a.k.a.  the multiplier. Finally, n refers to the element’s position in the series. For example, when n = 6 this refers to the 6^th element in the series from left to right. 

Next, I learned about the ‘sigma notation’ and ‘sum formula’ for a geometric series:

Form what I can tell, sigma notation tells you all the information about the series without actually telling you the value of the series (i.e. it tells you how many elements there are (n), which is the first element that is being used to make the calculation (a), etc.), whereas the sum formula is what you use to find the actual value of the geometric series at the n^th element. Sal went through a few videos on how to derive the sum formula which requires using a system-of-equations calculation. I understand how to do it but I don’t have the energy to write it out or explain it step by step.

I lastly learned about the same type of notation/formulas used for arithmetic series:

The only letter used in arithmetic series notation that’s not used in the geometric series notation is the letter d which represents the value being added to each successive element. For example, in the series {2 + 5 + 8 + … + n}, d = 3. Once again, the sum formula for arithmetic series is also derived using a system-of-equations calculation which, again, I understand but am not going to go through since I feel like garbage right now.

The Series unit test had 13 questions and I felt good about my understanding of every question except for the first one. In it I was asked to find the 1^st element’s value of a geometric series containing 10 elements and was given the sum of the series and the common ratio. I had to use the geometric sum formula in reverse (so to speak) which I managed to do but it took me a bit of trial and error. I was positive I got the correct answer in the end as I put my answer back into the formula which produced the sum I was initially given but I didn’t really know what I was doing throughout the question. Other than that, I felt good about everything else that came up.

This coming week I’ll be starting the unit Conic Sections (700/1700 M.P.) which I assume has to do with calculating the area and volume of cones. I’m hoping to get through this unit by the end of the week which, if I do, will be my last unit with M.P. to get through in the course. There’s one more unit remaining however, Probability and Combinatorics (800/800 M.P.), that, even though I already finished it, would probably be a good idea for me to review. I think there’s a reasonable chance that I can get through both units and the course test by the end of the next 2 weeks which is what I’m aiming for. Then finally (FINALLY) I’ll be starting calculus. 🤓