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Chapter 9: Problem 22

Write the first four terms of the sequence \(\left\\{a_{n}\right\\}\) defined by the following recurrence relations. $$a_{n+1}=a_{n}+a_{n-1} ; \quad a_{1}=1, a_{0}=1$$

Short Answer

Expert verified
Answer: The first four terms of the sequence are \(a_0 = 1\), \(a_1 = 1\), \(a_2 = 2\), and \(a_3 = 3\).

Step by step solution

01

Understand the Given Information

We are given the recurrence relation \(a_{n+1} = a_{n} + a_{n-1}\), with the initial conditions \(a_1 = 1\) and \(a_0 = 1\). This means we can find the next term of the sequence by adding the previous two terms.
02

Calculate the Second Term

To find the second term (\(a_2\)), we will use the given relation and the given initial conditions: $$a_2 = a_1 + a_0 = 1 + 1 = 2$$ So, the second term of the sequence is 2.
03

Calculate the Third Term

Now that we have the second term, we can find the third term by using the relation: $$a_3 = a_2 + a_1 = 2 + 1 = 3$$ So, the third term of the sequence is 3.
04

Calculate the Fourth Term

Finally, we will find the fourth term using the same method: $$a_4 = a_3 + a_2 = 3 + 2 = 5$$ So, the fourth term of the sequence is 5.
05

Summary

The first four terms of the sequence \(\left\\{a_{n}\right\\}\) defined by the recurrence relation \(a_{n+1} = a_{n} + a_{n-1}\) are: $$a_0 = 1, \quad a_1 = 1, \quad a_2 = 2, \quad a_3 = 3, \quad a_4 = 5.$$

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