Precalculus and Trigonometry · 35 minutes
Suggested grades 11–12 · Unit 3: Periodic models and finite sequences · Lesson 2 of 2
Finite Geometric Sums in a Design
Derive a finite-sum formula by subtracting shifted copies rather than adding every term.
Mr. S Teaches · Original starter lesson · HS-PRE-06 · Grade range is planning guidance.Watch, notice, discuss
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Formula for a finite geometric series | Sequences, series and induction | Precalculus | Khan Academy
By Khan Academy
Follow the finite-sum formula and connect the cancellation argument to the lesson’s design.
Open on YouTube: Formula for a finite geometric series | Sequences, series and induction | Precalculus | Khan Academy (opens a new tab) ↗Worked example: finite geometric series (sigma notation) | High School Math | Khan Academy
By Khan Academy
Count the terms carefully when translating sigma notation into a finite sum.
Open on YouTube: Worked example: finite geometric series (sigma notation) | High School Math | Khan Academy (opens a new tab) ↗How many terms are included, and how can you verify the sum without treating it as an infinite series?
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Learning objective
Derive and apply a finite geometric-series formula with a clearly counted number of terms.
Before you begin
- Geometric sequences
- Exponent rules
Gather your materials
- Paper and pencil
- Graph paper
- Calculator for checking results
Language to know
- finite series
- A sum containing a specified number of terms.
- common ratio
- The multiplier between consecutive geometric terms.
- index
- A label identifying a term’s position.
Understand the concept
If S=a+ar+…+ar^(n−1), then rS=ar+…+ar^n. Subtraction cancels the shared interior terms: S−rS=a−ar^n. For r≠1, S=a(1−r^n)/(1−r). If r=1, every term equals a and the sum is na. Counting terms carefully avoids using an exponent one too large.
Teach it step by step
- Write S=2+6+18+54 and 3S=6+18+54+162 on aligned lines.
- Subtract to obtain−2S=−160 and solve S=80.
- Generalize the cancellation to n terms and explicitly separate the case r=1.
- Apply the formula to a fictional layered pattern with five layers 2,4,8,16,32 tiles.
Make the thinking visible
Worked example or model response
For five layers, a=2,r=2,n=5: S=2(1−32)/(1−2)=62 tiles. Direct addition agrees. This is a finite total; no statement about an endless process is needed.
Guided practice
Find 1+1/2+1/4+1/8+1/16, showing the shifted-sum method.
Independent practice
1. Find the sum of six terms starting at 3 with ratio 2. 2. Find the sum of eight terms all equal 7. 3. Explain why r=1 cannot be substituted into the divided formula.
Quick assessment
For terms indexed 0 through 4, how many terms are being added?
Answer key & assessment criteria
Answers and teaching guidance
- Guided:31/16=1.9375; subtract half the sum to obtain(1/2)S=1−1/32.
- Independent 1:3(2⁶−1)=189. 2:56. 3: The denominator becomes zero, so use S=na.
- Quick check: Five terms; an initial index 0 does not mean there are only four.
Look for: Require the learner to explain the interpretation and check, not only state a result. Distinguish an exact value from a rounded approximation and attend to the stated domain.
Make it work for your learner
More support
Align two short numerical sums with matching terms colored before using symbols.
Extension challenge
Derive a finite arithmetic sum by pairing first and last terms and compare the reasoning.
Verified references · Selected contributions
Standards connections
Subject-content standards
These records identify the part of a subject standard practiced here. Course placement is instructional guidance; the source’s category or band remains authoritative.
Contributes to Common Core standard
HSA.SSE.B.4
Common Core · High school · Algebra
Regular high school mathematics standard
Derive and apply the formula for a finite geometric sum.
Read the official standard ↗This lesson contributes selected skills; broader applications and full standard coverage require further instruction.
Verified 2026-11-04. Selected contribution, not full attainment.
Questions & replies
Lesson discussion
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