When solving the non-homogeneous recurrence a_n = 2a_{n-1} + 2ⁿ by the method of undetermined coefficients, why must the trial particular solution be A·n·2ⁿ rather than simply A·2ⁿ?
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Incorrect.
The correct answer is C) Because 2 is a root of the homogeneous characteristic equation x = 2, so plain A·2ⁿ would be absorbed into the homogeneous solution and cancel to 0 = 2ⁿ — multiplying by n yields an independent form
Correct Answer
Because 2 is a root of the homogeneous characteristic equation x = 2, so plain A·2ⁿ would be absorbed into the homogeneous solution and cancel to 0 = 2ⁿ — multiplying by n yields an independent form
The associated homogeneous recurrence a_n = 2a_{n-1} has characteristic root 2, with general solution A·2ⁿ. Since the forcing term 2ⁿ shares this base ("resonance"), substituting the naive trial A·2ⁿ makes both sides cancel identically, giving no information. Multiplying by n (trial A·n·2ⁿ) breaks the degeneracy and yields a valid particular solution — the standard resonance rule for linear non-homogeneous recurrences and differential equations alike.