Problem 13 - Algebra for College Students

Question

Find the sum of each series. $$\sum_{i=1}^{10} 5 i^{0}$$

Step-by-Step Solution

Answer

50

Step 1: Understand the Summation Formula

The given exercise is a summation problem. We need to find the sum of the series \(\sum_{i=1}^{10} 5 i^{0}\). Here, \(i\) ranges from 1 to 10, and \(i^0\) means that for each term, the value of \(i\) is raised to the power of 0.

Step 2: Simplify the Expression Inside the Summation

Since \(i^0 = 1\) for any value of \(i\), each term in the series becomes \(5 \cdot 1 = 5\). Thus, the summation can be rewritten as \[\sum_{i=1}^{10} 5\cdot1= \sum_{i=1}^{10} 5\].

Step 3: Apply Summation Rule for Constants

We can factor out the constant 5 from the summation: \[ \sum_{i=1}^{10} 5 = 5 \sum_{i=1}^{10} 1 \].

Step 4: Calculate the Summation of 1

The summation \(\sum_{i=1}^{10} 1\) simply counts the number of terms from 1 to 10. There are 10 terms, therefore \ \sum_{i=1}^{10} 1= 10\.

Step 5: Multiply the Constant

Finally, multiply the constant 5 by the sum of the terms: \[ 5 \times 10 = 50 \]. Therefore, the sum of the series \( \sum_{i=1}^{10} 5 i^{0} \) is 50.

Key Concepts

Series Summation

Series summation involves adding a sequence of terms. In math, this is often represented using the summation symbol \(\sum\). This symbol indicates that you should add up all the values from a specified starting point to an ending point. For example, in the expression \(\sum_{i=1}^{10} 5 i^{0}\), the variable \(\it i\)\ changes from 1 to 10. Each term is then evaluated and summed together.
Understanding series summation is crucial in many areas of mathematics, including calculus, statistics, and even computer science.
Breaking down the problem, as done in the exercise we have, helps in comprehending each step taken to reach the final sum.

Summation of Constants

Summation of constants is a simpler form of series summation. A constant is a term that does not change, regardless of the index. For example, consider \(\sum_{i=1}^{10} 5 y^{0}\). Here, the term 5 is a constant.
The important rule to remember is that you can factor out this constant from the summation, as done in the exercise. Once the constant is outside the summation, the inner part \(\sum_{i=1}^{10} 1\) just counts the number of terms. In this case, from 1 to 10, there are 10 terms.
Therefore, the summation of constants simplifies many problems and speeds up calculations.

Index Notation

Index notation is a way to succinctly express series and sequences using a variable, often represented as \( i \) or sometimes \( n \). This variable changes over a defined range. In our exercise, \(\sum_{i=1}^{10} 5 y^0\), the index \(\it i\)\ runs from 1 to 10.
For each value of \( i \), the expression inside the summation symbol is evaluated next. In this case, the \( 5 i^0 \) becomes 5 times 1.
Index notation helps in keeping summation problems concise and manageable. It's a powerful tool in mathematical communication and problem solving.