Quick questions on Arithmetic sequences and series for HSC Maths Advanced financial modelling

An arithmetic sequence has a constant common difference $d$ between consecutive terms. With first term $a$,

Simple interest is charged only on the original principal $P$, so each period adds the same fixed amount $I = P r$ (with $r$ the per-period rate as a decimal). The totals $P + Pr, P + 2Pr, P + 3Pr, \dots$ form an arithmetic sequence with common difference $Pr$. The amount after $n$ periods is

An asset that loses a fixed dollar amount $D$ each year (the "prime cost" or straight-line method) has values $V_0, V_0 - D, V_0 - 2D, \dots$, an arithmetic sequence with common difference $-D$. The value after $n$ years is $V_n = V_0 - Dn$. Contrast this with declining-balance depreciation, which loses a fixed percentage each year and is therefore geometric.

A constant difference (you add $d$) is arithmetic; a constant ratio (you multiply by $r$) is geometric. Simple interest and straight-line depreciation are arithmetic; compound interest and declining-balance depreciation are geometric.

In straight-line depreciation the asset loses a fixed dollar amount, so $d$ is a subtraction of dollars, not a percentage of the current value. A fixed $1800 per year gives $d = -1800$, full stop.

When solving $S_n = X$ you get a quadratic; $n$ must be a whole number of terms. If the root is not an integer, check the terms either side rather than quoting a fraction.