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Grade 11 · Geometric sums

Geometric sums: A fractional starting term

Geometric sums: investigate a fractional starting term with first term a = 1.5; ratio r = 0.4.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

Paused
Geometric sums: A fractional starting term. Terms included: 1. Finite sum: 1.5. Last term: 1.5. Infinite limit: 2.5Finite sums and their limiting value02.51713partial sumterm count n → · labeled axes rescale to this model
The first term is positive and −1<r<1. Negative ratios are allowed. The limit is a/(1−r), including r=0, where the first term already equals the full sum. This is an exact finite sequence, not a random simulation.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Terms included
1
Finite sum
1.5
Last term
1.5
Infinite limit
2.5

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Understand what you are seeing

The idea behind the motion.

A common ratio generates each term from the previous one. Positive ratios give positive shrinking additions; negative ratios alternate signs and make partial sums approach the limit from alternating sides. The finite-sum formula and the infinite limit are different quantities. This investigation starts with First term a = 1.5; Ratio r = 0.4. Predict the result before playing, then change one parameter while keeping the other fixed to test the reason for the change.

A relationship to keep

Sₙ=a(1−rⁿ)/(1−r); |r|<1

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Set up the mathematical model

    Inspect the first term and the sign of the ratio before predicting the direction of the next change. The starting case is “A fractional starting term.”

  2. STEP 2

    Follow the changing quantity

    Play from one through thirteen terms. The dots are discrete partial sums, and connecting segments only guide the eye.

  3. STEP 3

    Explain and test the result

    Compare the last included term with the remaining tail. A small final term alone is not a general proof of convergence; here |r|<1 supplies the theorem.

Your turn to explain

Make a prediction. Test your reasoning.

Keep First term a = 1.5; Ratio r = 0.4. Pause the timeline at 80%. Given terms included = 10, calculate finite sum, last term, infinite limit. Show the substitution into the displayed formula.

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

For n=10, Sₙ=1.5(1−(0.4)^10)/(1−(0.4))=2.499738. The last included term is 1.5(0.4)^9=0.000393. The infinite limit comes from a/(1−r)=2.5. Results: Finite sum: 2.5; Last term: 0; Infinite limit: 2.5. Decimal values are rounded; retain the original parameters when checking.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Grade 11 · Algebra

Add repeated multiplicative contributions efficiently

Derive a finite geometric-sum formula by subtraction.