KKhaemenes PrecalculusTeacher Answer Key
Student Lesson

Teacher Key · Unit 10 · Lesson 05

Sigma Notation and Series Manipulation

Use this key to verify method, notation, domain, units, and interpretation. Accept equivalent exact forms when mathematically valid.

Khaemenes Academy · Teacher Key

Sigma Notation and Series Manipulation

U10 · L05
1

Find 3+7+11+15+19.

Series visualAnalyze sums with sigma notation, geometric convergence, binomial structure, and discrete financial or dynamical models. xy
Answer: 55
Reasoning: Five-term arithmetic sum.
2

Find the sum of the first 8 terms of 2,6,18,…

Series visualAnalyze sums with sigma notation, geometric convergence, binomial structure, and discrete financial or dynamical models. xy
Answer: 6560
Reasoning: Use S_n=a₁(r^n−1)/(r−1).
3

Does 12+6+3+… converge?

Series visualAnalyze sums with sigma notation, geometric convergence, binomial structure, and discrete financial or dynamical models. xy
Answer: Yes, to 24
Reasoning: r=1/2, so S∞=12/(1−1/2).
4

Evaluate Σ_{k=1}^{4} k².

Series visualAnalyze sums with sigma notation, geometric convergence, binomial structure, and discrete financial or dynamical models. xy
Answer: 30
Reasoning: 1+4+9+16=30.
5

Find the coefficient of x² in (1+x)^5.

Series visualAnalyze sums with sigma notation, geometric convergence, binomial structure, and discrete financial or dynamical models. xy
Answer: 10
Reasoning: Binomial coefficient C(5,2)=10.
6

State the convergence condition for an infinite geometric series.

Series visualAnalyze sums with sigma notation, geometric convergence, binomial structure, and discrete financial or dynamical models. xy
Answer: |r|<1
Reasoning: Only then do terms shrink to zero fast enough.
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