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Chapter 13: Problem 54

Rewrite the sums using sigma notation. $$5+5^{2}+5^{3}+\dots+5^{n}$$

Short Answer

Expert verified
The series can be rewritten in sigma notation as \( \sum_{i=1}^{n} 5^i \).

Step by step solution

01

Understanding the Series

First, recognize the pattern in the series. The series given is an exponential series where each term is a power of 5: \(5^1 + 5^2 + 5^3 + \, \cdots \, + 5^n\).

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Key Concepts

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

Exponential Series
An exponential series is a sequence of numbers where each term in the series is generated by raising a fixed base number to increasing powers, usually starting from 1 or 0. In the series that we are dealing with, the base number is 5, and each term is a power of 5: \( 5^1, 5^2, 5^3, \ldots \).
Each subsequent term is obtained by multiplying the previous term by the base, 5. This kind of series is fundamental in understanding geometric progressions and exponential growth.
  • Every term in the sequence relies on the previous one and is another power level higher.
  • This pattern helps when trying to sum the series or express it using specialized notation.
Overall, recognizing such exponential patterns is crucial in mathematics because it simplifies complex calculations and aids in predicting or expressing larger numbers.
Power Series
A power series is a type of infinite series that can represent a function as the sum of terms raised to powers. While our original exercise isn't about an infinite series, the concepts overlap.
In our case, the term power series can be partially applied because each term of the sequence in the exercise can be seen as a power of 5.
  • In mathematics, a power series generally takes the form \( a_0 + a_1x + a_2x^2 + a_3x^3 + \ldots \), where \( a_n \) are coefficients, and \( x \) is a variable raised to increasing powers.
  • Power series are used in calculus for approximating functions and solving differential equations.
  • It's essential not to confuse this with polynomial expressions, although they share similarities.
Recognizing terms as part of a power series improves understanding of their behavior and allows for manipulation and application in different mathematical contexts.
Mathematical Notation
Mathematical notation is a system of symbols and signs used to represent numbers and operations concisely. In the original exercise, we were asked to rewrite the exponential series using sigma notation.
Sigma notation, represented by the Greek letter \( \Sigma \), is a way to express the sum of a sequence of terms. Its typical form is:
  • \( \Sigma_{i=m}^{n} a_i \), where \( i \) is the index of summation, \( m \) is the starting index, \( n \) is the ending index, and \( a_i \) is the general term of the sequence.
  • In our case, since each term follows \( 5^i \), the series can be rewritten as: \( \Sigma_{i=1}^{n} 5^i \).
  • This notation makes it clear how many terms are added and what the general form of each term is.
Understanding mathematical notation is crucial because it provides a universal language for describing complex ideas succinctly. It also simplifies the process of evaluating expressions and communicating mathematical concepts effectively.

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Most popular questions from this chapter

The sum of three consecutive terms in an arithmetic sequence is \(6,\) and the sum of their cubes is \(132 .\) Find the three terms.

Use a calculator to complete Compute the fifth roots of \(i .\) Express your answers in rectangular form, with the real and imaginary parts rounded to two decimal places.

Use DeMoivre's theorem to find the indicated roots. Express the results in rectangular form. Square roots of \(-\frac{1}{2}-\frac{1}{2} \sqrt{3} i\)

Find the indicated roots. Express the results in rectangular form. In the identity \((\cos \theta+i \sin \theta)^{2}=\cos 2 \theta+i \sin 2 \theta,\) carry out the multiplication on the left-hand side of the equation. Then equate the corresponding real parts and imaginary parts from each side of the equation that results. What do you obtain?

Carry out the indicated expansions. $$\left(4 A-\frac{1}{2}\right)^{5}$$
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