/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 3 Suppose the sequence \(\left\\{a... [FREE SOLUTION] | 91影视

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Chapter 9: 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
Question: Find the first five terms of the sequence given by the recurrence relation \(a_{n+1} = n a_{n}\), with \(a_1 = 1\). 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

Determine the first term of the sequence

We are given the first term of the sequence, \(a_1 = 1\).
02

Use the recurrence relation to find the second term of the sequence

We can use the recurrence relation \(a_{n+1} = n a_n\) to find the second term of the sequence. For \(n = 1\), we have: \(a_{1+1} = 1 a_{1} \Rightarrow a_2 = 1 \cdot 1 \Rightarrow a_2 = 1\) Thus, the second term of the sequence is \(a_2 = 1\).
03

Use the recurrence relation to find the third term of the sequence

Next, we find the third term by using the recurrence relation with \(n = 2\): \(a_{2+1} = 2 a_{2} \Rightarrow a_3 = 2 \cdot 1 \Rightarrow a_3 = 2\) Thus, the third term of the sequence is \(a_3 = 2\).
04

Use the recurrence relation to find the fourth term of the sequence

Now we find the fourth term by using the recurrence relation with \(n = 3\): \(a_{3+1} = 3 a_{3} \Rightarrow a_4 = 3 \cdot 2 \Rightarrow a_4 = 6\) Thus, the fourth term of the sequence is \(a_4 = 6\).
05

Use the recurrence relation to find the fifth term of the sequence

Finally, we find the fifth term by using the recurrence relation with \(n = 4\): \(a_{4+1} = 4 a_{4} \Rightarrow a_5 = 4 \cdot 6 \Rightarrow a_5 = 24\) Thus, the fifth term of the sequence is \(a_5 = 24\). In conclusion, the first five terms of the sequence \(\{a_n\}\) are: $$a_1 = 1, a_2 = 1, a_3 = 2, a_4 = 6, a_5 = 24$$

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