We just finished our 2nd full week of PreCalc 11 and I’m very glad to say that for the first time, I genuinely understand the concept of everything we learned. Our first unit is “Sequences and Series,” and today we finished lesson 1.4. Basically we learned about arithmetic and geometric series and sequences. I really love how there is a specific equation for each lesson and if we plug in the right numbers, it is pretty straightforward. My favourite thing about it is writing everything out and when it all pieces together, it’s very satisfying.
Here are a few things I learned this week (with examples at the bottom):
What makes a sequence arithmetic is if the pattern of numbers have the same common difference, you should be able to take a term add or subtract it by the common difference, then take that term and do the same, then continue on like that. It’s important to have a rule for this sequence so that if you needed to find a really large term, you could use the equation, plug in the right numbers and find that number. The equation for arithmetic sequences is: = + (n-1)d. stands for the first term, (n-1) stands for the term you’re looking for, minus 1, and d stands for the common difference.
Arithmetic Series are a bit different; a series is simply just the total sum of terms in a sequence. If the series is 2, 4, 6, 8, 10… the series would just be 2+4+6+8+10… The equation to find the Arithmetic Series is… = ( – ) or if you don’t have you can also use this equation, it just has more steps… = (2 (n-1)d).
A geometric sequence is almost the same as Arithmetic sequence but instead of having a common difference where you add or subtract a certain number each time to get the next term, you have something called a common ration and you just multiply (or divide) a designated number every time. To find the common ratio, you just have to have two terms, take the second term (on the right of the first term) and divide it by the term first listed that will give you the common ratio. Now, to find larger terms, you need an equation and the equation is… +x
The most recent thing we learned is geometric series, the equation for this one is the longest but I find that if you write it down quite a bit, you’ll catch on very fast. Just like arithmetic and geometric sequences, arithmetic and geometric series are almost alike, the only difference is that geometric series is the total sum of a geometric sequence. The equation to find a term in a geometric series would be: = a(1-) ÷ 1-r
Something that I think that can best explain my understanding of these few lessons is to show how I solve a few equations.