/*! 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 36 Find each indicated sum. $$\su... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 36

Find each indicated sum. $$\sum_{i=2}^{4}\left(-\frac{1}{3}\right)^{i}$$

Short Answer

Expert verified
The sum is approximately -0.46296

Step by step solution

01

Identify the Rule and Range

Look at the sigma notation and identify the rule \(\left(-\frac{1}{3}\right)^{i}\) and the range from 2 to 4.
02

Apply the Rule

Apply the rule to each value of \(i\) in the range. It will be \(\left(-\frac{1}{3}\right)^{2}\), \(\left(-\frac{1}{3}\right)^{3}\), \(\left(-\frac{1}{3}\right)^{4}\). Calculate these three terms.
03

Sum up the terms

Add the three terms calculated in the previous step to get the sum: \[\left(-\frac{1}{3}\right)^{2} + \left(-\frac{1}{3}\right)^{3} + \left(-\frac{1}{3}\right)^{4}\]

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Key Concepts

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

Summation
Summation is a mathematical operation that allows you to add a sequence of numbers. It's represented by the Greek letter sigma (\( \Sigma \)).
In our exercise, summation notation is used to add a sequence of terms generated by the rule \( \left(-\frac{1}{3}\right)^{i} \).
The notation \( \sum_{i=2}^{4} \) indicates that you need to calculate the expression for each integer \( i \) from 2 to 4 and then sum those results.
  • First, substitute each \( i \) in the rule \( \left(-\frac{1}{3}\right)^{i} \): for \( i = 2, 3, 4 \).
  • Next, compute each term: \( \left(-\frac{1}{3}\right)^{2} \), \( \left(-\frac{1}{3}\right)^{3} \), and \( \left(-\frac{1}{3}\right)^{4} \).
  • Finally, add the results of these computations together to find the total sum.
The summation provides a concise way to deal with patterns and sequences in math.
Negative Fractions
Negative fractions might seem tricky at first, but they're just fractions with a negative sign.
In our exercise, we are working with the fraction \( -\frac{1}{3} \).
This indicates that the fraction is less than zero. Understanding how to handle negative fractions is crucial, especially when they involve exponents.
  • The negative sign can appear in various places in a fraction: before the fraction, in the numerator, or in the denominator. All represent the same value.
  • When multiplying a negative fraction by itself (like in our rule \( \left(-\frac{1}{3}\right)^{i} \)), the result alternates between positive and negative based on the exponent.
Using negative fractions requires careful attention to signs, particularly when combined with other mathematical operations like exponents.
Exponents
Exponents are a way to express repeated multiplication of a number by itself.
In our case, the rule \( \left(-\frac{1}{3}\right)^{i} \) involves raising the fraction \( -\frac{1}{3} \) to the power of \( i \).
Let's break it down:
  • An exponent tells you how many times to multiply the base (in this case, \( -\frac{1}{3} \)) by itself.
  • If \( i \) is even, the multiplication results in a positive fraction.
  • If \( i \) is odd, the result retains the negative sign of the base.
Understanding exponents helps solve complex problems quickly, especially when dealing with sequences or series. The interplay of exponents with other operations, such as summation and dealing with negative signs, is key to mastering these concepts.

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Most popular questions from this chapter

Show that $$ 1+2+3+\cdots+n=\frac{n(n+1)}{2} $$ is true for the given value of \(n .\) $$n=3: \text { Show that } 1+2+3=\frac{3(3+1)}{2}$$

In Exercises \(39-44\), you are dealt one card from a 52 -card deck. Find the probability that you are dealt a 7 or a red card.

In a class of 50 students, 29 are Democrats, 11 are business majors, and 5 of the business majors are Democrats. If one student is randomly selected from the class, find the probability of choosing a. a Democrat who is not a business major. b. a student who is neither a Democrat nor a business major.

This will help you prepare for the material covered in the next section. Use the formula \(a_{n}=a_{1} 3^{n-1}\) to find the seventh term of the sequence \(11,33,99,297, \ldots\)

Write an equation in point-slope form and slope-intercept form for the line passing through \((-2,-6)\) and perpendicular to the line whose equation is \(x-3 y+9=0 .\) (Section 1.5 Example 2 )
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