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Chapter 10: Problem 13

Explicit formulas Write the first four terms of the sequence \(\left {a_{n}\right\\}_{n=1}^{\infty}\) $$a_{n}=\frac{1}{10^{n}}$$

Short Answer

Expert verified
Question: Determine the first four terms of the sequence with an explicit formula given as \(a_{n}=\frac{1}{10^{n}}\). Answer: The first four terms of the sequence are \(0.1, 0.01, 0.001,\) and \(0.0001\).

Step by step solution

01

Substitute the values of n into the explicit formula

Replace \(n\) with the numbers \(1\) to \(4\) in the formula \(a_{n}=\frac{1}{10^{n}}\).
02

Calculate the first term (n=1)

Substitute \(n=1\) into the formula: \(a_{1}=\frac{1}{10^{1}}=\frac{1}{10}=0.1\)
03

Calculate the second term (n=2)

Substitute \(n=2\) into the formula: \(a_{2}=\frac{1}{10^{2}}=\frac{1}{100}=0.01\)
04

Calculate the third term (n=3)

Substitute \(n=3\) into the formula: \(a_{3}=\frac{1}{10^{3}}=\frac{1}{1000}=0.001\)
05

Calculate the fourth term (n=4)

Substitute \(n=4\) into the formula: \(a_{4}=\frac{1}{10^{4}}=\frac{1}{10000}=0.0001\)
06

Write down the first four terms of the sequence

The first four terms of the sequence \(\left\\{a_{n}\right\\}_{n=1}^{\infty}\) with explicit formula \(a_{n}=\frac{1}{10^{n}}\) are: \(0.1, 0.01, 0.001,\) and \(0.0001\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sequences
Sequences are ordered lists of numbers that follow a particular pattern or rule. They are fundamental concepts in mathematics, helping us understand patterns and structures. A sequence can be thought of as a list of numbers that are presented in a specific order. Each number has a particular position or rank.
In the example given, the sequence is defined by the explicit formula \(a_{n} = \frac{1}{10^{n}}\). This formula tells us that each term of the sequence can be found by dividing 1 by 10 raised to the power of the term's position \(n\).
When dealing with sequences, keep in mind:
  • Terms in a sequence are numbered, usually starting from 1.
  • An explicit formula is used to determine the general term of the sequence.
  • Sequences can be finite or infinite, in this example, it is infinite.
Mathematical Calculation
Mathematical calculation is the process of computing an answer or solving a problem using a mathematical method. In the context of sequences, calculations are performed to determine specific terms based on the given formula.
In our sequence example, calculations involve substituting the term number into the sequence's explicit formula and simplifying:
  • For \(n=1\), \(a_1 = \frac{1}{10^1} = 0.1\)
  • For \(n=2\), \(a_2 = \frac{1}{10^2} = 0.01\)
  • For \(n=3\), \(a_3 = \frac{1}{10^3} = 0.001\)
  • For \(n=4\), \(a_4 = \frac{1}{10^4} = 0.0001\)
By performing calculations, we determine each term in the sequence. This is essential for understanding the sequence's behavior and identifying any patterns.
Terms of a Sequence
The terms of a sequence are the individual elements or numbers in the sequence. Each term is associated with a specific position denoted usually by an integer \(n\).
The position of each term is important since it helps define the pattern of the sequence. For example, in the provided sequence \(a_{n} = \frac{1}{10^n}\):
  • The first term \(a_1\) is 0.1, which is derived when \(n = 1\).
  • The second term \(a_2\) is 0.01, from \(n = 2\).
  • The third term \(a_3\) is 0.001, which results when \(n = 3\).
  • The fourth term \(a_4\) is 0.0001, found by setting \(n = 4\).
The first few terms of a sequence can often reveal a lot about its pattern and behavior. Identifying the terms helps in understanding and solving problems.
Substitution Method
The substitution method is a technique used to solve equations or find values by replacing one variable with another. When finding terms in a sequence with an explicit formula, the substitution method is essential.
Here's how it works in sequences:
  • Identify the explicit formula, such as \(a_{n}=\frac{1}{10^{n}}\).
  • Choose a value for \(n\), representing the position of the term you want to find.
  • Substitute this value of \(n\) into the formula to calculate the corresponding term.
By using the substitution method, you efficiently determine specific terms of a sequence. It highlights how the formula changes with different inputs, showcasing the sequence's unique pattern or rule.

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Most popular questions from this chapter

Infinite products An infinite product \(P=a_{1} a_{2} a_{3} \ldots .,\) which is denoted \(\prod_{k=1}^{\infty} a_{k},\) is the limit of the sequence of partial products \(\left\\{a_{1}, a_{1} a_{2}, a_{1} a_{2} a_{3}, \ldots\right\\} .\) Assume \(a_{k} > 0\) for all \(k\) and \(L\) is a finite constant. a. Evaluate \(\prod_{k=1}^{\infty}\left(\frac{k}{k+1}\right)=\frac{1}{2} \cdot \frac{2}{3} \cdot \frac{3}{4} \cdot \frac{4}{5} \cdots\) b. Show that if \(\sum_{k=1}^{\infty} \ln a_{k}=L,\) then \(\prod_{k=1}^{\infty} a_{k}=e^{L}\) c. Use the result of part (b) to evaluate \(\prod_{k=0}^{\infty} e^{1 / 2^{t}}=e \cdot e^{1 / 2} \cdot e^{1 / 4} \cdot e^{1 / 8} \cdots\)

Archimedes' quadrature of the parabola The Greeks solved several calculus problems almost 2000 years before the discovery of calculus. One example is Archimedes' calculation of the area of the region \(R\) bounded by a segment of a parabola, which he did using the "method of exhaustion." As shown in the figure, the idea was to fill \(R\) with an infinite sequence of triangles. Archimedes began with an isosceles triangle inscribed in the parabola, with area \(A_{1},\) and proceeded in stages, with the number of new triangles doubling at each stage. He was able to show (the key to the solution) that at each stage, the area of a new triangle is \(1 / 8\) of the area of a triangle at the previous stage; for example, \(A_{2}=1 / 8 A_{1}\) and so forth. Show, as Archimedes did, that the area of \(R\) is \(4 / 3\) times the area of \(A_{1}\).

Estimate the value of the following convergent series with an absolute error less than \(10^{-3} .\) $$\sum_{k=1}^{\infty} \frac{(-1)^{k}}{k^{5}}$$

Compare the growth rates of \(\left\\{n^{100}\right\\}\) and \(\left\\{e^{n / 100}\right\\}\) as \(n \rightarrow \infty\).

The expression $$\begin{aligned} &1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+}}}}\\\ &\end{aligned}$$ where the process continues indefinitely, is called a continued fraction. a. Show that this expression can be built in steps using the recurrence relation \(a_{0}=1, a_{n+1}=1+\frac{1}{a_{n}}\) for \(n=0,1,2,3, \ldots\) Explain why the value of the expression can be interpreted as \(\lim _{n \rightarrow \infty} a_{n},\) provided the limit exists. b. Evaluate the first five terms of the sequence \(\left\\{a_{n}\right\\}\) c. Using computation and/or graphing, estimate the limit of the sequence. d. Assuming the limit exists, use the method of Example 5 to determine the limit exactly. Compare your estimate with \(\frac{1+\sqrt{5}}{2},\) a number known as the golden mean. e. Assuming the limit exists, use the same ideas to determine the value of $$\begin{aligned} &a+\frac{b}{a+\frac{b}{a+\frac{b}{a+\frac{b}{a+}}}}\\\ &\end{aligned}$$ where \(a\) and \(b\) are positive real numbers.
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