Today's soundtrack is Jesus Piece: Only Self, a 30-minute technical hardcore masterpiece that borders on deathcore, kind of like what I think a collaboration between Hatebreed and "The Unspoken King" era Cryptopsy might sound like. Really good stuff, with lots of memorable moments. For example, I love the tongue-in-cheek Six Feet Under-esque breakdown on "Dog No Longer," and the funeral doom style of "II," very reminiscent of Ahab or Mournful Congregation.
This morning, I'm working on geometric sequences. Like arithmetic sequences, a geometric sequence is a sequence of numbers; however, instead of adding a constant to a number to create the sequence, "[a] geometric sequence is formed by multiplying each term after the 1st term by a constant, to determine the next term" (Pearson, Pre-Calculus 11, p. 30). Whereas in arithmetic sequences we have the common difference, d, in geometric sequences, we refer to the constant as a common ratio, r, which must be "any non-zero real number" (p. 30).
There are two ways that we categorize geometric sequences: whether they are finite or infinite, and whether they are convergent or divergent.
We can easily determine whether a geometric sequence is finite or infinite; the latter is shown by an ellipsis after the last listed term.
If a geometric sequence has a positive common ratio and moves consistently toward zero from either direction, that sequence is convergent. If it has a negative common ratio or moves away from zero in either direction, it is divergent.
A geometric sequence will always work with this equation: t1, t1r, t1r², t1r³, ... t1rⁿ⁻¹. The names of the preceding terms are t1, t2, t3, t4, ... tn. We see here that the term number is always one more than the term's exponent number.
To find the common ratio of a geometric sequence when we know the first and second terms, we divide the second term by the first; the result is the common ratio.
The book gives as an example this sequence: "512, -256, 128, -64, ..." (p. 31). So we would say that r = -256/512, which is equal to r = -1/2, or r = -0.5.
To find the common ratio of a geometric sequence where we know t1 and tn, we use the general term equation: tn = t1(r)ⁿ⁻¹, and solve for r.
To find the common ratio of a geometric sequence where we know at least two values of tns, let tx equal the lower known term value, let ty equal the higher known term number; let n1 equal the higher known number’s term, let n2 equal the lower known value’s term. Using the equation tx(r)ⁿ¹⁻ⁿ²=ty, we solve for r. (I figured this out out myself; it took a while!)
To find the value of t1, we use the equation tn = t1(r)ⁿ⁻¹ and solve for t1.
To determine the number of terms in a finite sequence when we know the value of t1, the common ratio, and the final term, we use the general term equation tn = t1(r)ⁿ⁻¹, input our numbers, and solve for n.
To find the value of tn, we first determine the common ratio, then use the general term equation tn = t1(r)ⁿ⁻¹, solving for tn.