Find 3+7+11+15+19.
Answer: 55
Reasoning: Five-term arithmetic sum.
Reasoning: Five-term arithmetic sum.
Teacher Key · Unit 10 · Lesson 02
Arithmetic Sequences and Finite Series
Use this key to verify method, notation, domain, units, and interpretation. Accept equivalent exact forms when mathematically valid.
Khaemenes Academy · Teacher Key
Arithmetic Sequences and Finite Series
Find the sum of the first 8 terms of 2,6,18,…
Answer: 6560
Reasoning: Use S_n=a₁(r^n−1)/(r−1).
Reasoning: Use S_n=a₁(r^n−1)/(r−1).
Does 12+6+3+… converge?
Answer: Yes, to 24
Reasoning: r=1/2, so S∞=12/(1−1/2).
Reasoning: r=1/2, so S∞=12/(1−1/2).
Evaluate Σ_{k=1}^{4} k².
Answer: 30
Reasoning: 1+4+9+16=30.
Reasoning: 1+4+9+16=30.
Find the coefficient of x² in (1+x)^5.
Answer: 10
Reasoning: Binomial coefficient C(5,2)=10.
Reasoning: Binomial coefficient C(5,2)=10.
State the convergence condition for an infinite geometric series.
Answer: |r|<1
Reasoning: Only then do terms shrink to zero fast enough.
Reasoning: Only then do terms shrink to zero fast enough.