Geometric Series – the Proof
About this app
Sₙ = 1 + q + … + qⁿ⁻¹ as bars: times q, one slot along, subtract – the middle cancels, only qⁿ and 1 remain. Six scrubbable steps to Sₙ = (qⁿ − 1)/(q − 1), the case q = 1, q < 1 and a savings-plan link.
With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.
The sum formula of the geometric series
The finite geometric series is Sₙ = 1 + q + q² + … + qⁿ⁻¹, the sum of the first n terms of a geometric sequence. For q ≠ 1, Sₙ = (qⁿ − 1)/(q − 1); for q = 1, Sₙ = n. The proof fits on one line: multiplying Sₙ by q gives q + q² + … + qⁿ. Subtracting Sₙ cancels every term from q to qⁿ⁻¹, leaving q·Sₙ − Sₙ = qⁿ − 1, so Sₙ(q − 1) = qⁿ − 1. In financial mathematics q = 1 + p/100 is the interest factor.
What you see
The terms stand side by side as bars, each q times as tall as the one before. In six steps every bar is multiplied by q and moves one slot along, the new series q·Sₙ is placed above Sₙ, equal pairs cancel, and only the ends qⁿ and 1 remain. A second card lays the terms end to end as lengths: the remainder qⁿ − 1, stretched by 1/(q − 1), is exactly as long as Sₙ. The working updates with the numbers you set.
Try it
Drag a bar up or down to change q between 0.5 and 2, and drag the handle to get up to twelve terms. With q = 1 you see why the formula fails there: both ends are equally tall, the remainder is zero, and you cannot divide by q − 1 = 0. With q = 0.8 the remainder and the denominator turn negative, and the strip flips when dividing. The example Saving 5 % links to a savings plan: ten yearly payments of €1,000 at year end grow to €12,577.89.
Frequently asked questions
- How do you prove the geometric series sum formula?
- Write Sₙ = 1 + q + … + qⁿ⁻¹ and q·Sₙ = q + q² + … + qⁿ one below the other and subtract. All terms from q to qⁿ⁻¹ cancel, leaving q·Sₙ − Sₙ = qⁿ − 1. Factoring gives Sₙ(q − 1) = qⁿ − 1, and for q ≠ 1 dividing yields Sₙ = (qⁿ − 1)/(q − 1).
- Why does the formula not work for q = 1?
- For q = 1 the denominator q − 1 is zero. The equation Sₙ(q − 1) = qⁿ − 1 becomes 0 = 0 and says nothing about Sₙ. The sum can be counted directly instead: every term is 1, so Sₙ = n.
- Where is the geometric series used in financial mathematics?
- Savings plans, annuities and loan repayment compound payments over several years with the interest factor q = 1 + p/100. Paying an amount E at the end of each year for n years gives a final balance of E·Sₙ = E·(qⁿ − 1)/(q − 1); paying at the start of each year adds one more factor q.
- What can I set in the app?
- You choose q between 0.5 and 2 and n between 1 and 12, with sliders or directly on the bars. The proof runs in six steps that you can select one by one or scrub with the timeline slider. Examples for growth, saving at 5 %, decline and q = 1 are ready to use.
Subject: Finance
More from Finance
- Annuity Loan – Equal Instalments
- Savings Formula – Fixed Instalments
- Arithmetic vs. Geometric
- Loan types compared
- Perpetuity & Drawdown
- Sub-annual Payments & Equivalent Annual Amount
- Compounding Frequency & Effective Rate
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