KKhaemenes PrecalculusOpen-Age · Printable Analytical Lesson
Unit 10Course HomeTeacher Key

Unit 10 · Lesson 01 · Week 29 · Sequences, Series & Discrete Growth

Sequences, Recursive Definitions & Explicit Rules

Model discrete change with explicit and recursive sequences, arithmetic and geometric patterns, and finite or infinite series. In this lesson, the central focus is sequences, recursive definitions & explicit rules. Learners move among symbolic, graphical, numerical, and contextual evidence rather than relying on one representation alone.

Open-AgePrintableFoundation · Core · ExtendedMastery target 80%

Learning intentions

  • Interpret the defining quantities and notation used in sequences, recursive definitions & explicit rules.
  • Construct or analyze at least two representations related to sequences, recursive definitions & explicit rules.
  • Show symbolic work with correct domains, units, intervals, or angle measures.
  • Explain why the selected method is valid and check the result against another representation.

Essential question

How do recursive and explicit structures describe discrete change, accumulation, convergence, and financial or scientific processes?

Multiple-representation model

Sequences visualModel discrete change with explicit and recursive sequences, arithmetic and geometric patterns, and finite or infinite series. xy

Model discrete change with explicit and recursive sequences, arithmetic and geometric patterns, and finite or infinite series.

a_n=a_1+(n-1)da_n=a_1r^{n-1}S_n=\frac n2(2a_1+(n-1)d)

Worked reasoning

Explicit ruleAn arithmetic sequence 7,12,17,… has a_n=7+5(n−1).
ConvergenceFor 18+6+2+…, r=1/3, so S∞=18/(1−1/3)=27.

Misconception check: A recursive rule needs an initial value; without it, the sequence is not uniquely determined.

Communication and reasoning

Write a complete response that identifies the representation used, shows the symbolic steps, interprets parameters or units, and verifies the conclusion with a second representation or reasonableness check.

Khaemenes Academy · Open-Age Precalculus

Sequences, Recursive Definitions & Explicit Rules

Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and calculator evidence when used.

U10 · L01
Name
Date
1

Find the next term: 4,9,14,19,…

Sequences visualModel discrete change with explicit and recursive sequences, arithmetic and geometric patterns, and finite or infinite series. xy
2

Write an explicit rule for 4,9,14,…

Sequences visualModel discrete change with explicit and recursive sequences, arithmetic and geometric patterns, and finite or infinite series. xy
3

Find a_8 for a_n=3·2^{n−1}.

Sequences visualModel discrete change with explicit and recursive sequences, arithmetic and geometric patterns, and finite or infinite series. xy
4

Give a recursive rule for 10,7,4,1,…

Sequences visualModel discrete change with explicit and recursive sequences, arithmetic and geometric patterns, and finite or infinite series. xy
5

Is 2,6,18,54 arithmetic or geometric?

Sequences visualModel discrete change with explicit and recursive sequences, arithmetic and geometric patterns, and finite or infinite series. xy
6

Find the 20th term of a_n=7+4(n−1).

Sequences visualModel discrete change with explicit and recursive sequences, arithmetic and geometric patterns, and finite or infinite series. xy
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