/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 43 Find the sum for each series. $$... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 43

Find the sum for each series. $$\sum_{i=1}^{5}\left(3^{i}-4\right)$$

Short Answer

Expert verified
The sum of the series is 343.

Step by step solution

01

Identify the Series Terms

The series given is \( \sum_{i=1}^{5} (3^i - 4) \). We need to identify each term from \( i = 1 \) to \( i = 5 \). These terms are: \( 3^1 - 4 \), \( 3^2 - 4 \), \( 3^3 - 4 \), \( 3^4 - 4 \), and \( 3^5 - 4 \).
02

Calculate Each Term in the Series

Now, calculate each individual term:- For \( i = 1 \), \( 3^1 - 4 = 3 - 4 = -1 \).- For \( i = 2 \), \( 3^2 - 4 = 9 - 4 = 5 \).- For \( i = 3 \), \( 3^3 - 4 = 27 - 4 = 23 \).- For \( i = 4 \), \( 3^4 - 4 = 81 - 4 = 77 \).- For \( i = 5 \), \( 3^5 - 4 = 243 - 4 = 239 \).
03

Add All Terms to Find the Sum

Add up all the calculated terms from Step 2:\(-1 + 5 + 23 + 77 + 239\).Calculate the sum:\(-1 + 5 = 4\)\(4 + 23 = 27\)\(27 + 77 = 104\)\(104 + 239 = 343\)

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Key Concepts

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

Arithmetic Series
An arithmetic series is a series of numbers in which each term increases by a constant difference. These series are particularly important in the study of mathematics because they allow us to calculate long sequences quickly.
  • For example, the series 2, 4, 6, 8 is an arithmetic series where each number increases by 2.
  • The nth term can be calculated using the formula: \( a_n = a + (n-1) \cdot d \), where \( a \) is the first term and \( d \) is the common difference.
In the exercise, the given series is not an arithmetic series but rather involves exponential functions subtracted by a constant. Understanding arithmetic series, however, provides a foundation for recognizing different types of series we might encounter.
Exponential Functions
Exponential functions form a cornerstone of mathematics, often written as \( f(x) = a^x \) where \( a \) is a constant and \( x \) is an exponent. These functions grow at increasingly rapid rates as the exponent increases. In our exercise, the terms in the series include the exponential function \( 3^i \), where the base 3 is repeatedly raised to successive powers from 1 to 5.
  • Exponential growth occurs when the rate of change of a quantity is proportional to the current amount of that quantity.
  • Examples are population growth, bank interest calculations, and radioactive decay.
By incorporating an exponential function in the series, we see how quickly the terms grow. For instance, in the step-by-step solution, we identified the exponential growth from 3, 9, 27, 81 to 243 as the power increases from 1 to 5.
Algebraic Expressions
Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They are fundamental in forming equations and solving problems.
  • An example of an algebraic expression is \( 3^i - 4 \), where \( 3^i \) is the exponential expression and \( -4 \) is a constant term.
  • Algebraic expressions can be simplified or manipulated to find unknown values by applying algebraic techniques.
In the exercise, each term in the series is an algebraic expression composed of exponential and constant parts. Understanding these expressions is crucial for calculating the overall sum of the series, as demonstrated when we calculated each term individually and summed them up.

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Most popular questions from this chapter

What will happen when an infectious disease is introduced into a family? Suppose a family has \(I\) infected members and \(S\) members who are not infected, but are susceptible to contracting the disease. The probability \(P\) of \(k\) people not contracting the disease during a 1-week period can be calculated by the formula. $$P=\left(\begin{array}{l}S \\\k\end{array}\right) q^{k}(1-q)^{S-k}$$ where \(q=(1-p)^{l}\) and \(p\) is the probability that a susceptible person contracts the disease from an infectious person. For example, if \(p=0.5,\) then there is a \(50 \%\) chance that a susceptible person exposed to one infectious person for 1 week will contract the disease. (Source: Hoppensteadt, F. and C. Peskin, Mathematics in Medicine and the Life Sciences, Springer-Verlag.) (a) Approximate the probability \(P\) of 3 family members not becoming infected within 1 week if there are currently 2 infected and 4 susceptible members. Assume that \(p=0.1\) (b) A highly infectious disease can have \(p=0.5 .\) Repeat part (a) with this value of \(p\) (c) Approximate the probability that everyone would become sick in a large family if initially \(I=1, S=9\) and \(p=0.5\)

Solve each problem . (Modeling) Investment for Retirement According to T. Rowe Price Associates, a person who has a moderate investment strategy and \(n\) years until retirement should have accumulated savings of \(a_{n}\) percent of his or her annual salary. The geometric sequence $$ a_{n}=1276(0.916)^{n} $$ gives the appropriate percent for each year \(n\) (a) Find \(a_{1}\) and \(r\) (b) Find and interpret the terms \(a_{10}\) and \(a_{20}\)

Prove each statement for positive integers \(n\) and \(r,\) with \(r \leq n\) (Hint: Use the definitions of permutations and combinations.) $$C(n, n-1)=n$$

Prove each statement for positive integers \(n\) and \(r,\) with \(r \leq n\) (Hint: Use the definitions of permutations and combinations.) $$C(n, n)=1$$

Find \(a_{1}\) for each arithmetic sequence. $$a_{6}=-8, a_{7}=-18$$
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