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Chapter 8: Problem 3

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

Expert verified
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

01

Find the initial term a鈧�

The given initial term is \(a_1=1\).
02

Find the second term a鈧�

Use the recurrence relation \(a_{n+1} = n a_n\). For the second term, set \(n=1\): \(a_{2} = 1 a_{1} = 1 \cdot 1 = 1\).
03

Find the third term a鈧�

Use the recurrence relation \(a_{n+1} = n a_n\). For the third term, set \(n=2\): \(a_{3} = 2 a_{2} = 2 \cdot 1 = 2\).
04

Find the fourth term a鈧�

Use the recurrence relation \(a_{n+1} = n a_n\). For the fourth term, set \(n=3\): \(a_{4} = 3 a_{3} = 3 \cdot 2 = 6\).
05

Find the fifth term a鈧�

Use the recurrence relation \(a_{n+1} = n a_n\). For the fifth term, set \(n=4\): \(a_{5} = 4 a_{4} = 4 \cdot 6 = 24\).
06

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 \(a_{5} = 24\).

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