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Chapter 19: Problem 4

A sequence is given by \(5, \frac{5}{8}, \frac{5}{27}, \frac{5}{64}, \ldots\) Write down an expression to denote the full sequence.

Short Answer

Expert verified
Answer: The general expression for the given sequence is \(\frac{5}{n^3}\), where n is the position of the term in the sequence.

Step by step solution

01

Identify the pattern in the sequence

Observe the given sequence: \(5, \frac{5}{8}, \frac{5}{27}, \frac{5}{64}, \ldots\). Notice that each term's numerator is 5, and the denominators are increasing in powers of 2, 3, 4, and so on. Therefore, the sequence can be rewritten as: \(5, \frac{5}{2^3}, \frac{5}{3^3}, \frac{5}{4^3}, \ldots\).
02

Find the general expression for the sequence

From the pattern identified in Step 1, we can determine the general expression for the nth term using the following format: \(\frac{5}{n^3}\), where n is the position of the term in the sequence.
03

Write down the expression for the full sequence

Using the general expression from Step 2, we can write down the expression to denote the full sequence as: \(\left\{\frac{5}{n^3}\right\}_{n=1}^{\infty}\). This means the sequence consists of terms where the numerator is 5 and the denominator is the cube of each natural number n, starting from 1 to infinity.

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Most popular questions from this chapter

Explain carefully the distinction between a sequence and a series.

Find the Maclaurin expansion for \(\sin ^{2} x\). (Hint: use a trigonometrical identity and the series for \(\sin x\).)

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