Problem

For \(2,~5,~10,~17\), compute the first differences and express their odd-number pattern as a formula in \(n\). Give a one-based recurrence beginning with \(a_{1}=2\), include \(n\ge~2\), and regenerate all listed terms.

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The recurrence is driven by the changing first differences, so their pattern must be expressed using the destination index. We’ll compute the consecutive increases, match them to an odd-number formula that starts correctly at the second term, write the initial condition and index domain, and regenerate every listed value.

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

a₁ = 2 and, for n ≥ 2, aₙ = aₙ₋₁ + (2n − 1). The differences are 3, 5, 7, so it generates a₂ = 5, a₃ = 10, and a₄ = 17.

a₁ = 2 and, for n ≥ 2, aₙ = aₙ₋₁ + (2n + 1). This gives a₂ = 7, not 5.

a₁ = 2 and, for n ≥ 1, aₙ = aₙ₋₁ + (2n + 3). This gives a₂ = 9, not 5, and mixes zero-based indexing with a₁.

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For \(2,~5,~10,~17\), compute the first differences and express their odd-number pattern as a formula in \(n\). Give a one-based recurrence beginning with \(a_{1}=2\), include \(n\ge~2\), and regenerate all listed terms.

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