/*! 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 34 \(\sum_{i=3}^{10} 5=\) (A) 260... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 34

\(\sum_{i=3}^{10} 5=\) (A) 260 (B) 50 (C) 40 (D) 5 (E) none of these

Short Answer

Expert verified
C) 40

Step by step solution

01

Understanding the Problem

The given expression is a summation from \(i=3\) to \(i=10\) of the term 5. It basically means to add the number 5 eight times, where \(i\) is the index that varies from 3 to 10.
02

Perform the Summation

In this case, since each term in the series is equal (each term is 5), this summation can be calculated by simple multiplication. The total sum can be computed by multiplying the number of terms (10-3+1=8 terms) with each term (5), i.e., \(8 * 5 = 40\).
03

Identify the Answer

The result of the sum is 40.

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Key Concepts

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

Summation Notation
Summation notation is a mathematical shorthand used for expressing the concept of adding consecutive elements in a series. It's represented with the Greek letter sigma (\(\Sigma\)). Here鈥檚 a breakdown of its components:
  • The symbol itself \(\Sigma\) indicates that a sum needs to be taken.

  • Below and above the \(\Sigma\), you usually find an expression for the range of the index, such as \(i = 3\) to \(i = 10\) in our example.

  • Beside the \(\Sigma\), you鈥檒l see the expression that needs to be summed; here, it's simply the number 5, meaning each item in the series is the constant 5.
This notation allows you to neatly express calculations involving repeated additions, especially helpful for both small and large series. Understanding how this works is key to performing arithmetic summations efficiently.
Index of Summation
The index of summation is a crucial component in the use of summation notation. This index generally is a dummy variable that takes on integer values over a defined range, driving how many and which terms are summed.
  • In our example, the index is \(i\), and it outlines the beginning and ending terms of the series - here from \(3\) to \(10\).

  • This index helps in counting the number of terms present in the series. In this example, values \(3, 4, 5, 6, 7, 8, 9,\) and \(10\) result in 8 terms total.
It鈥檚 important to remember that the index itself doesn鈥檛 impact the values summed, but instead indicates the number of times the expression needs to be repeated, ensuring that every term required to be summed is included in your total calculation.
Constant Series
A constant series is a sequence in which each term is exactly the same. When dealing with summation notation with a constant, the simplicity factor kicks in.
  • In the given exercise, each term in the sequence is a constant 5, so adding them all simply involves multiplying this constant by the number of times it appears, which is determined by the index range.

  • Here, we multiply 5 by 8 (since 8 terms are in the series from \(i=3\) to \(i=10\)), directly giving us the result of 40.
The advantage of a constant series is that it shortens the computational process to basic multiplication, which is more efficient than performing separate additions for each term. This is particularly useful when handling larger ranges where the constant remains the same throughout the entire series.

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