Savings Formula – Fixed Instalments
About this app
A savings plan with a constant instalment: each instalment as a tower on its payment date, the interest staircase and little Gauss n(n+1)/2 for simple interest, the geometric series Kₙ = E·q·(qⁿ−1)/(q−1) with compounding, in advance and in arrears, balance, present value and building savings.
With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.
The savings formula
With a fixed yearly instalment E, the final balance is the sum of all separately accumulated instalments. With q = 1 + p/100, instalment E_k earns interest for n − k + 1 years in advance (start of year) and n − k years in arrears (end of year). Compound interest gives the geometric series Kₙ = E · q · (qⁿ − 1)/(q − 1); in arrears the factor q drops out. With simple interest each instalment earns the same piece E · p/100 per year, so Kₙ = n · E + E · p/100 · n(n + 1)/2: the little Gauss formula. The present value B₀ = Kₙ/qⁿ is the single deposit that reaches the same balance.
What you see
The top card shows each instalment as a tower on its payment date; its height is its final value: the deposit at the bottom, the interest pieces above as a staircase, the interest on interest on top. The oldest instalment is the tallest tower. A copy of the staircase rotated by 180° completes it to an n × (n + 1) rectangle. The bottom card traces the account balance as a step curve next to the curve without compounding. When you collect, the towers fly into a total stack sorted into n · E, simple interest and compound interest. The Working panel puts all numbers into the lecture formulas.
Try it
Set the instalment (100 to 10,000 €), the interest rate (0.5 to 12 %) and the term (1 to 30 years), and switch between simple and compound interest and between payment in advance and in arrears. In advance the final balance is higher by exactly the factor q. Drag a tower to change the instalment, or drag the top of the total stack to a savings goal and the app computes the required instalment. The building savings example loads €40,000 after 6 years at 3 % and gives an instalment of €6,183.90.
Frequently asked questions
- What is the savings plan formula for payments in advance and in arrears?
- With q = 1 + p/100, instalment E and n years, payments in advance give Kₙ = E · q · (qⁿ − 1)/(q − 1) and payments in arrears K̂ₙ = E · (qⁿ − 1)/(q − 1). In advance every instalment earns one more year of interest, so Kₙ = q · K̂ₙ. For €1,000 a year at 5 % over 10 years this gives €13,206.79 and €12,577.89.
- What does the little Gauss formula have to do with a savings plan?
- With simple interest every instalment earns the same interest piece E · p/100 per year. The oldest instalment collects n pieces, the next n − 1 and so on, in total 1 + 2 + … + n = n(n + 1)/2 pieces. So Kₙ = n · E + E · p/100 · n(n + 1)/2, or (n − 1)n/2 for payments in arrears.
- What is the present value of a savings plan?
- The present value B₀ is the amount you would have to invest once at the start to reach the same final capital with compound interest: B₀ = Kₙ/qⁿ. For €1,000 a year paid in advance at 5 % over 10 years it is €8,107.82, in arrears €7,721.73.
- What can I adjust in this app?
- You set instalment, interest rate and term, choose simple or compound interest and payment in advance or in arrears. You can drag the towers and the total stack directly, complete the Gauss staircase to a rectangle, show the present value and load the building savings example. The app uses yearly instalments with interest credited at the end of each year.
Subject: Finance
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