Today's soundtrack is Breaking Benjamin: Ember, a hard rock album that ticks all the boxes for me. It has heavy guitars, the drums sound huge, and the vocals are alternately singing/screaming. The production is very sterile, almost over-produced, but in this case, it works. The album reminds me of a cross between Project 86, Pillar, and Demon Hunter, in the best possible way.
This morning, I'm putting together my formula sheet to prepare for my test on arithmetic and geometric sequences and series.
Formulas for Arithmetic Sequence, Arithmetic Series, Geometric Sequence, Geometric Series, and Infinite Geometric Series
- - General formula: tn = t1 + d (n - 1)
- - General formula 1: Sn = n (t1 + tn) / 2
- - General formula 2: Sn = n[2t1 + d(n - 1)] / 2 (Which one we use depends on whether we are provided with d and/or tn!)
- - General formula: tn = t1(r)ⁿ⁻¹
- - Find common ratio when first two terms known: r = t2/t1
- - Find common ratio when two other terms known: 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, and solve for r in the equation tx(r)ⁿ¹⁻ⁿ²=ty
- - General formula: Sn = t1(1-[r]ⁿ)/1-r, r≠1
- - General formula: S∞ = t1 / 1-r
Legend (taken from "Formulas for Sequences and Series, Pre-Calculus 11"):
tn = nth term
t1 = first term
n = number of terms
d = common difference
Sn = sum of first n terms
r = common ratio
S∞ = sum of infinite number of terms