Suppose the sequence { aₙ} is defined by the recurrence relation a₍ₙ₊₁₎ = n · aₙ , for n=1, 2, 3 ...., where a₁ = 1. Write out the first five terms of the sequence.

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Identify the given recurrence relation: \(a_{n+1} = n \cdot a_n\) for \(n = 1, 2, 3, \ldots\), with the initial term \(a_1 = 1\).

Calculate the second term \(a_2\) by substituting \(n=1\) into the recurrence: \(a_2 = 1 \cdot a_1\).

Calculate the third term \(a_3\) by substituting \(n=2\): \(a_3 = 2 \cdot a_2\).

Calculate the fourth term \(a_4\) by substituting \(n=3\): \(a_4 = 3 \cdot a_3\).

Calculate the fifth term \(a_5\) by substituting \(n=4\): \(a_5 = 4 \cdot a_4\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Recurrence Relations

A recurrence relation defines each term of a sequence using previous terms. Understanding how to apply the given formula step-by-step is essential to generate terms of the sequence from initial values.

Sequence Terms Calculation

Calculating terms involves substituting values of n into the recurrence relation and using previously found terms. This process helps in explicitly writing out the first few terms of the sequence.

Factorials and Growth Patterns

The given recurrence resembles factorial growth since each term multiplies the previous term by n. Recognizing this pattern aids in understanding the behavior and magnitude of the sequence terms.