/*! 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 16 Evaluate. $$ \sum_{i=1}^{4} ... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 16

Evaluate. $$ \sum_{i=1}^{4} 3^{i-1} $$

Short Answer

Expert verified
The sum is 40.

Step by step solution

01

Understanding the Sigma Notation

Sigma notation, \( \sum_{i=1}^{4} 3^{i-1} \), is a way to represent a sum where \( i \) starts at 1 and ends at 4. For each value of \( i \), we substitute into \( 3^{i-1} \) and add the results.
02

Substitute and Calculate for Each Term

We will evaluate each term by substituting \( i \) from 1 to 4 into the expression \( 3^{i-1} \). - For \( i=1 \), it's \( 3^{1-1} = 3^0 = 1 \).- For \( i=2 \), it's \( 3^{2-1} = 3^1 = 3 \).- For \( i=3 \), it's \( 3^{3-1} = 3^2 = 9 \).- For \( i=4 \), it's \( 3^{4-1} = 3^3 = 27 \).
03

Sum All Calculated Values

Now, we add the calculated values from each term:\( 1 + 3 + 9 + 27 = 40 \).

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Key Concepts

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

Exponential Expressions
Exponential expressions are a way to denote repeated multiplication of a number by itself. These expressions are written with two main components: a base and an exponent. The exponent tells us how many times to multiply the base by itself. For example, in the expression \(3^4\), 3 is the base, and 4 is the exponent, indicating that we should multiply 3 by itself four times: \(3 \times 3 \times 3 \times 3\).
  • If the exponent is 0, any nonzero base raised to this power equals 1. This is why \(3^0 = 1\) in our exercise.
  • An exponent of 1 simply indicates the base itself, so \(3^1 = 3\).
  • Higher exponents imply further repeated multiplication, like \(3^2 = 9\) and \(3^3 = 27\).
Exponential expressions are commonly used in computing growth patterns, sciences, finance, and more.
Series Evaluation
Series evaluation involves calculating the sum of a sequence of terms. When you see sigma notation, such as \(\sum_{i=1}^{4} 3^{i-1}\), it signifies that we will sum a series of terms based on some formula. Here's how it works:
  • The symbol \(\sum\) represents the process of summation.
  • The expression under the sum tells us what formula to use for creating the sequence.
  • The range of numbers below and above the sigma, \(i=1\) to 4, tells us the bounds for the index \(i\).
In the series evaluation, we calculated each term by varying \(i\) from 1 to 4 and using the expression \(3^{i-1}\) for each value. Then, we summed up all the resulting numbers: 1, 3, 9, and 27, reaching the final result of 40. This step-by-step summation is fundamental to understanding how sequences behave and how results accumulate.
Mathematical Notation
Mathematical notation is a collection of symbols used to convey mathematical ideas and concepts succinctly and efficiently. In our exercise, sigma notation \(\sum\) plays this role by providing a concise way to represent the addition of a series of numbers. Here鈥檚 how it breaks down for this problem:
  • The symbol \(\sum\) is the summation sign, indicating that a sum of terms follows.
  • The limits of summation, \(i=1\) to 4, specify a "loop" where \(i\) takes each integer value from 1 through 4.
  • The expression \(3^{i-1}\) is the formula used for each term in the series, depending on the value of \(i\).
Understanding mathematical notation is crucial because it allows us to clearly and precisely convey complex mathematical operations and ideas without lengthy explanations. Having proficiency in reading and interpreting these symbols is key to solving mathematical problems efficiently.

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Most popular questions from this chapter

Find partial sum. Find the sum of the first four terms of the sequence whose general term is \(a_{n}=-2 n\).

Use the partial sum formula to find the partial sum of the given arithmetic or geometric sequence. See Examples 1 and 4 Find the sum of the first eight terms of the geometric sequence \(-1,2,-4, \dots\)

Solve. The spraying of a field with insecticide killed 6400 weevils the first day, 1600 the second day, 400 the third day, and so on. Find the total number of weevils killed during the first 5 days.

Write series with summation notation. \(4+12+36+108\)

Evaluate. \(\frac{\frac{6}{11}}{1-\frac{1}{10}}\)
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