Grade 11 · Geometric sums · 129 of 600
Geometric sums · First term a=3.5; Ratio r=-0.2
Include the first n=8 terms, from ar⁰ through ar⁷. Calculate the finite sum, last included term, and infinite-series limit. Givens: First term a=3.5; Ratio r=-0.2.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
A geometric sequence starts at 3.5 and has common ratio r=-0.2. Include the first n=8 terms, from ar⁰ through ar⁷. Calculate the finite sum, last included term, and infinite-series limit.
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A starting point
Use a(1−r⁸)/(1−r) for eight terms. Because |r|<1, the infinite sum is a/(1−r).
Work through the reasoning
Step 1
Identify the model and target
A geometric sequence starts at 3.5 and has common ratio r=-0.2. Include the first n=8 terms, from ar⁰ through ar⁷. The governing relation is Sₙ=a(1−rⁿ)/(1−r); |r|<1. Use a(1−r⁸)/(1−r) for eight terms. Because |r|<1, the infinite sum is a/(1−r).
Step 2
Substitute and calculate
For n=8, Sₙ=3.5(1−(-0.2)^8)/(1−(-0.2))=2.916659. The last included term is 3.5(-0.2)^7=-0.000045. The infinite limit comes from a/(1−r)=2.916667.
Step 3
Check the mathematical meaning
The sum of the omitted tail is ar⁸/(1−r)=0.000007. Adding that tail to the finite sum recovers 2.916667. The last included exponent is 7, not 8. Terms included: 8; Finite sum: 2.917; Last term: -0; Infinite limit: 2.917. Decimal values are rounded, so use unrounded intermediate values.
The answer
For n=8, Sₙ=3.5(1−(-0.2)^8)/(1−(-0.2))=2.916659. The last included term is 3.5(-0.2)^7=-0.000045. The infinite limit comes from a/(1−r)=2.916667. The sum of the omitted tail is ar⁸/(1−r)=0.000007. Adding that tail to the finite sum recovers 2.916667. The last included exponent is 7, not 8. Animation check: Terms included: 8; Finite sum: 2.917; Last term: -0; Infinite limit: 2.917. Decimal displays are rounded; retain the original parameters and exact π until the final step.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
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Watch the relationship
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The mathematical idea
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. A geometric sequence starts at 3.5 and has common ratio r=-0.2. Include the first n=8 terms, from ar⁰ through ar⁷. The requested state occurs at 60% playback; use the exact target stated in the question for your calculation.
Sₙ=a(1−rⁿ)/(1−r); |r|<1
03 · Reflect and transfer
Explain what changes and why.
How does a negative common ratio change the side from which successive partial sums approach the limit?
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.