Sequences has been an interesting unit. I didn’t think I’d learn as many new things as I did, or improve. It seemed a very easy concept at first, and simple, but nothing in math is easy and simple if you delve deep enough into it. Because of this, I shall leave behind a beacon of hope for my future self that will surely be freaking out a few days prior to the final. And when fretting for a test I will remember to study all the formulas, but as I have learned, they only get you so far. In this light, I will describe the mechanics of each and tidbits I am sure to forget.

Arithmetic Sequences:

Arithmetic sequences are sequences that go up by a common difference “d” each term. This means that with the common difference and any term “t” in the entire string of terms “n” it is possible to find every term in the sequence. By looking at each number and subtracting any term to the direct left by its right side partner anywhere in the sequence, you can find “d”.

Geometric Sequences:

Geometric sequences are ones that increase by a multiplier “r”. Using any term from this variant of sequence combined with “r” makes finding every term possible. Often with geometric sequences finding the common ratio is a lot less straightforward than the common difference of arithmetic sequences. By dividing any term to the direct left its right side partner anywhere in the sequence, you can find “r”.

Sums:

Sums encompass both Arithmetic and Geometric Sequences. In our questions, they have been: adding up all of the terms to a certain key term down the line. For example, S_{50} would be the sum of all the terms up to the fiftieth term. In geometric sequences, a formula can be applied with less knowledge needed than arithmetic sums; S_n=(\frac{a(r^n-1)}{r-1}) to find the sum. The most critical information for solving this is finding t_1“a” and “r”. Without these it is almost always impossible to solve the sum. “n” is often given due to the number in subscript of the sum, however this isn’t always the case and paying attention to when the question gives you “r”, “n”, “a,” or a different set of terms (for example t5 and t7) for you to find “a” with, is critical. When given t_5 and t_7 (or any other combo that doesn’t include t_1) adding the smaller one with a hypothetical “r” will give you this. t_7=t_5+r^2. 2 is used with r because 7-5 is 2.

I hope my future fervent self will be calmed by this knowledge, and anyone else willing to give it a read. Goodbye, and good luck.