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Chapter 9: Problem 17

Write the first four terms of the sequence \(\left\\{a_{n}\right\\}\) defined by the following recurrence relations. $$a_{n+1}=2 a_{n} ; \quad a_{1}=2$$

Short Answer

Expert verified
Answer: The first four terms of the sequence are \(a_{1}=2\), \(a_{2}=4\), \(a_{3}=8\), and \(a_{4}=16\).

Step by step solution

01

Use the initial condition

The initial condition is given as \(a_{1} = 2\). This is the first term of the sequence.
02

Apply the recurrence relation to find the second term, \(a_2\)

Use the recurrence relation \(a_{n+1}=2 a_{n}\) and the initial condition \(a_{1}=2\). Substitute \(n = 1\) into the recurrence relation: $$a_{2} = 2 a_{1} = 2 \cdot 2 = 4$$
03

Apply the recurrence relation to find the third term, \(a_3\)

Again, use the recurrence relation with the value we found for \(a_2\): $$a_{3} = 2 a_{2} = 2 \cdot 4 = 8$$
04

Apply the recurrence relation to find the fourth term, \(a_4\)

Finally, use the recurrence relation with the value we found for \(a_3\): $$a_{4} = 2 a_{3} = 2 \cdot 8 = 16$$
05

Write down the first four terms of the sequence

We have found the first four terms of the sequence \(\left\\{a_{n}\right\\}\) using the initial condition and the recurrence relation: $$a_{1} = 2$$ $$a_{2} = 4$$ $$a_{3} = 8$$ $$a_{4} = 16$$

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Most popular questions from this chapter

a. Sketch the function \(f(x)=1 / x\) on the interval \([1, n+1]\) where \(n\) is a positive integer. Use this graph to verify that $$ \ln (n+1)<1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}<1+\ln n $$ b. Let \(S_{n}\) be the sum of the first \(n\) terms of the harmonic series, so part (a) says \(\ln (n+1)0,\) for \(n=1,2,3, \ldots\) c. Using a figure similar to that used in part (a), show that $$ \frac{1}{n+1}>\ln (n+2)-\ln (n+1) $$ d. Use parts (a) and (c) to show that \(\left\\{E_{n}\right\\}\) is an increasing sequence \(\left(E_{n+1}>E_{n}\right)\) e. Use part (a) to show that \(\left\\{E_{n}\right\\}\) is bounded above by 1 f. Conclude from parts (d) and (e) that \(\left\\{E_{n}\right\\}\) has a limit less than or equal to \(1 .\) This limit is known as Euler's constant and is denoted \(\gamma\) (the Greek lowercase gamma). g. By computing terms of \(\left\\{E_{n}\right\\},\) estimate the value of \(\gamma\) and compare it to the value \(\gamma \approx 0.5772 .\) (It has been conjectured, but not proved, that \(\gamma\) is irrational.) h. The preceding arguments show that the sum of the first \(n\) terms of the harmonic series satisfy \(S_{n} \approx 0.5772+\ln (n+1)\) How many terms must be summed for the sum to exceed \(10 ?\)

Evaluate the limit of the following sequences. $$a_{n}=\frac{4^{n}+5 n !}{n !+2^{n}}$$

Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than \(10^{-4} .\) Although you do not need it, the exact value of the series is given in each case. $$\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{(2 k+1) !}$$

A glimpse ahead to power series Use the Ratio Test to determine the values of \(x \geq 0\) for which each series converges. $$\sum_{k=0}^{\infty} x^{k}$$

Determine whether the following series converge absolutely or conditionally, or diverge. $$\sum_{k=1}^{\infty} \frac{(-1)^{k} \tan ^{-1} k}{k^{3}}$$
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