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Arithmetic vs. Geometric

About this app

Arithmetic and geometric sequences as bar pairs in three lecture scenarios: output with +d or +p %, straight-line vs. declining-balance depreciation, simple vs. compound interest. A difference bracket and a quotient bracket at the selected year, two type-test cards whose flat track reveals the type, slide exercises with solutions and a quiz. Refutes "declining-balance depreciation eventually reaches zero".

With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.

Constant difference or constant ratio

A sequence is arithmetic if the difference of consecutive terms is constant: aₙ = a + (n − 1) · d. It is geometric if the ratio is constant: aₙ = a · qⁿ⁻¹. The applications of financial mathematics are almost always one of the two: output with a fixed increase d or with p % per year (q = 1 + p/100), straight-line depreciation Bₙ = K − n · K/N or declining-balance depreciation Rₙ = K · qⁿ with q = 1 − p/100, simple interest Kₙ = K(1 + n · p/100) or compound interest Kₙ = K · qⁿ.

What the app shows

Each year has a pair of bars: the arithmetic sequence on the left, the geometric one on the right. The step to the previous year is marked. At the selected year one bracket measures the difference and a second one the ratio. Two cards beside it plot the difference and the ratio to the previous year across all years. For one sequence the difference is flat, for the other the ratio, and that reveals the type. A ring marks the break-even point; for depreciation a second ring marks where straight-line depreciation reaches zero after N years.

Try it yourself

Switch between output, depreciation and interest and drag the handles on the bars: the left one changes d or the useful life N, the right one the percentage p. The formulas appear with the numbers filled in, and the table lists values, differences and ratios. Load the lecture exercises: 15 % depreciation over 10 years (19.69 % remaining), the rate for 10 % remaining after 10 years (20.57 %), or how long €1000 at 20 % takes to drop below €100 (11 years). In the quiz you identify the type of number sequences.

Frequently asked questions

What is the difference between an arithmetic and a geometric sequence?
In an arithmetic sequence the same amount d is added from term to term, for example 3, 5, 7, 9 with d = 2. In a geometric sequence each term is multiplied by the same factor q, for example 3, 6, 12, 24 with q = 2. To check, form the differences and the ratios of neighbouring terms; whichever stays constant determines the type.
Why does declining-balance depreciation never reach zero?
With declining-balance depreciation a fixed percentage p of the remaining value is written off each year, so Rₙ = K · qⁿ with q = 1 − p/100. A power of a positive number is never zero, so the remaining value becomes arbitrarily small but stays positive. Straight-line depreciation Bₙ = K − n · K/N, by contrast, reaches zero after exactly N years.
How do you compute the depreciation rate for a target remaining value?
From R_N = K · q^N it follows that p = 100 · (1 − ᴺ√(R_N/K)). If an asset is to be written down to 10 % of its purchase value in 10 years, p = 100 · (1 − ¹⁰√0.1) ≈ 20.57 % per year. If p is given instead, N = (ln R_N − ln K) / ln(1 − p/100) gives the duration, rounded up to whole years.
What can I adjust in the app?
You choose the scenario output, depreciation or interest and set the initial value, the increase d or the useful life N, the percentage p, the range of years and the year n, with sliders or by dragging the bars. Five exercises from the lecture can be loaded with their solution. A quiz shows random sequences whose type you identify.

Subject: Finance

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