Question

Suppose the sequence \(\left\\{a_{n}\right\\}\) is defined by the recurrence relation \(a_{n+1}=n a_{n},\) for \(n=1,2,3, \ldots,\) where \(a_{1}=1 .\) Write out the first five terms of the sequence.

Short answer

Answer: The first five terms of the sequence are \(a_{1} = 1\), \(a_{2} = 1\), \(a_{3} = 2\), \(a_{4} = 6\), and \(a_{5} = 24\).

Step-by-step solution

Find the initial term a₁

The given initial term is \(a_1=1\).

Find the second term a₂

Use the recurrence relation \(a_{n+1} = n a_n\). For the second term, set \(n=1\): \(a_{2} = 1 a_{1} = 1 \cdot 1 = 1\).

Find the third term a₃

Use the recurrence relation \(a_{n+1} = n a_n\). For the third term, set \(n=2\): \(a_{3} = 2 a_{2} = 2 \cdot 1 = 2\).

Find the fourth term a₄

Use the recurrence relation \(a_{n+1} = n a_n\). For the fourth term, set \(n=3\): \(a_{4} = 3 a_{3} = 3 \cdot 2 = 6\).

Find the fifth term a₅

Use the recurrence relation \(a_{n+1} = n a_n\). For the fifth term, set \(n=4\): \(a_{5} = 4 a_{4} = 4 \cdot 6 = 24\).

Write out the first five terms of the sequence

We found the first five terms of the sequence according to the given recurrence relation: \(a_{1} = 1\), \(a_{2} = 1\), \(a_{3} = 2\), \(a_{4} = 6\), and \(a_{5} = 24\).