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F-BF.1.a Warmup M2-014-A10-V02

Build quadratic and exponential functions from context using explicit formulas, recursive processes, or calculation steps.

Write a recursive rule for a quadratic-looking sequence from its terms

Problem

Analyze the one-based sequence \(1,~4,~9,~16\). Find its first differences, represent the \(\text{nth}\) added amount for \(n\ge~2\), write the complete recurrence, and use it to reproduce \(4,~9,~\text{and}~16\).

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aₙ = aₙ₋₁ + (2n − 1). Without an initial term a₁, the recurrence cannot generate the sequence.

a₁ = 1 and, for n ≥ 1, aₙ = aₙ₋₁ + (2n + 3). This gives a₂ = 8, not 4, and mixes indexing systems.

a₁ = 1 and, for n ≥ 2, aₙ = aₙ₋₁ + (2n + 1). This gives a₂ = 6, not 4.

a₁ = 1 and, for n ≥ 2, aₙ = aₙ₋₁ + (2n − 1). The differences are 3, 5, 7, so it generates 4, 9, and 16.

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Curriculum context

Course: Math II
Standard: F-BF.1.a
Category: Functions
Domain: Building Functions
Objective: Build quadratic and exponential functions from context using explicit formulas, recursive processes, or calculation steps.
Problem type: Write a recursive rule for a quadratic-looking sequence from its terms