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Chapter 2: Problem 131

Write each sum in sigma notation. \(1+q+q^{2}+q^{3}+q^{4}+\cdots+q^{n-1}\)

Short Answer

Expert verified
The sum is expressed as \( \sum_{k=0}^{n-1} q^k \).

Step by step solution

01

Identify the Pattern

The given expression is a sum of terms where each term is a power of the variable \( q \). The terms are \( 1, q, q^2, q^3, \) and so on up to \( q^{n-1} \). The exponent increases by 1 for each subsequent term. This indicates an arithmetic progression of exponents, starting from 0 up to \( n-1 \).
02

Understand Sigma Notation

Sigma notation is a way to write the sum of a sequence using the Greek letter \( \Sigma \), which stands for 'sum'. In this case, the sequence is \( q^0, q^1, q^2, \ldots, q^{n-1} \). We need to express this sequence using a general term and bounds for the summation.
03

Write the General Term

The general term of the sequence is \( q^k \), where \( k \) starts from 0 and ends at \( n-1 \). This represents each term in the sequence we identified earlier.
04

Define the Limits of Summation

The sigma notation needs limits for \( k \). Since the sequence starts at \( k = 0 \) and goes to \( k = n-1 \), these are our lower and upper bounds, respectively.
05

Combine into Sigma Notation

Put everything together in sigma notation: \[\sum_{k=0}^{n-1} q^k\]This compactly represents the sum of the sequence from \( q^0 \) to \( q^{n-1} \).

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Key Concepts

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

Arithmetic Progression
An arithmetic progression (AP) is a sequence of numbers in which the difference between consecutive terms is constant. However, in the given exercise, the focus is on the arithmetic progression of exponents rather than the numbers themselves.

In our sequence, the exponents are increasing by 1 each time. This is a key feature of an arithmetic progression:
  • The difference, known as the common difference, is consistent for each step. Here it is 1.
  • The sequence of exponents starts at 0 and goes up to \(n-1\).
  • Each term can be described by the function \(k\), where \(k\) varies from 0 to \(n-1\).
This pattern helps us translate a long list into a concise formula using sigma notation. Recognizing this is the first step in writing sums succinctly.
Exponents
Exponents are a shorthand used in mathematics to represent repeated multiplication. In our sequence, each term is a power of \(q\):
  • \(q^0\) equals 1, since any number to the power of 0 is 1.
  • \(q^1\) is simply \(q\) itself.
  • \(q^2, q^3, \) and so on represent multiplying \(q\) by itself for the given number of times.
The sequence \(1, q, q^2, q^3, \ldots, q^{n-1}\) therefore shows \(q\) being raised to increasing powers.

Understanding how exponents work is crucial, as they form the basis for expressing large and small numbers efficiently. They also make recognizing patterns easier, which is essential for translating expressions into sigma notation.
Summation Limits
In the context of sigma notation, the limits of summation define where the sequence starts and ends. For the given series, the bounds are:
  • The series starts at \(k = 0\). This corresponds to the first term, which is \(q^0\).
  • The series ends at \(k = n-1\), reaching up to \(q^{n-1}\).
These limits, \(0\) to \(n-1\), are written below and above the sigma (\(\Sigma\)) symbol. They tell us exactly how many terms are in the sum and how to evaluate it.

By setting these boundaries, sigma notation provides a compact way of expressing sums, making complex calculations easier to handle visually and computationally. Remember, the key is to make sure the general term reflects the actual pattern and that the limits accurately represent the start and end of the series.

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Most popular questions from this chapter

The sequence \(\mid a_{n}\) \\} is recursively defined. Compute \(a_{n}\) for \(n=1,2, \ldots, 5\) $$ a_{n+1}=1+2 a_{n}, a_{0}=0 $$

Model painkillers that are absorbed into the blood from a slow release pill. Ourmathematical model for the amount, \(a_{t}\), of drug in the blood t hours after the pill is taken must include the amount absorbed from the pill each hour. Our model starts with the word equation. $$ \begin{array}{c} a_{t+1}=a_{t}+\begin{array}{l} \text { amount absorbed } \\ \text { from the pill } \end{array}-\begin{array}{l} \text { amount eliminated } \\ \text { from the blood } \end{array} \end{array} $$ Assume the amount absorbed from the pill between time \(t\) and time \(t+1\) is \(20 \cdot(0.2)^{t}\). (a) The drug has first order elimination kinetics. \(40 \%\) of the drug is eliminated from the blood each hour. Write down the recursion relation for \(a_{t+1}\) in terms of \(a_{t}\) (b) Assuming that \(a_{0}=0\), meaning that no drug is present in the blood initially, calculate the amount of drug present at times \(t=1,2, \ldots, 6\) (c) What is the maximum amount of drug present at any time in this interval? At what time is this maximum amount reached? (d) Use a spreadsheet to calculate the amount of drug present in hourly intervals from \(t=0\) up to \(t=24\). (e) Show that, when \(t\) is large, the amount of drug present in the blood decreases approximately exponentially with \(t .\) Hint: Plot the values that you computed for \(a_{t}\) against \(t\) on semilogarithmic axes.

Because of complex interactions with other drugs, some drugs have zeroth order elimination kinetics in some circumstances, and first order kinetics in other circumstances, depending on what other drugs are in the patient's system, as well as on age and preexisting medical conditions. Use the data on how concentration varies with time to determine whether the drug has zeroth or first order kinetics. Given the following sequence of measurements of drug concentration, determine whether the drug has zeroth or first order kinetics. $$ \begin{array}{lcccc} \hline \boldsymbol{t} \text { (Hours) } & 1 & 2 & 3 & 4 \\ \hline c_{t}(\mu \mathrm{g} / \mathrm{ml}) & 40 & 36 & 32 & 28 \\ \hline \end{array} $$

Assume that the population growth is described by the Beverton-Holt recruitment curve with parameters \(R_{0}\) and a. Find the population sizes for \(t=1,2, \ldots, 5\) and find \(\lim _{t \rightarrow \infty} N_{t}\) for the given initial value \(N_{0} .\) \(R_{0}=3, a=1 / 20, N_{0}=7\)

Use the formal definition of limits to show that \(\lim _{n \rightarrow \infty} a_{n}=a ;\) that is, find \(N\) such that for every \(\epsilon>0\), there exists an \(N\) such that \(\left|a_{n}-a\right|<\epsilon\) whenever \(n>N\). $$ \lim _{n \rightarrow \infty} \frac{1}{n^{2}}=0 $$
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