Problem 29 - Algebra and Trigonometry

Question

find each indicated sum. $$ \sum_{i=1}^{6} 5 i $$

Step-by-Step Solution

Answer

The sum for the given exercise is 105.

Step 1: Identify the function and the range

The given function is 5i and the range of the summation is from i=1 to i=6.

Step 2: Apply the summation formula

The sum of the first 'n' natural numbers is given by \( \frac{n(n+1)}{2} \). Substituting n = 6 into this formula gives \( \frac{6 * (6+1)}{2} = 21 \).

Step 3: Apply the given function

Next, multiply the summation value by 5 (as per the function 5i). This gives a final result of \( 21 * 5 = 105 \).

Key Concepts

Arithmetic Series

An arithmetic series is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the 'common difference'. For example, in the sequence 2, 4, 6, 8, the common difference is 2. An arithmetic series takes the form of adding all these terms together. If you have the first term, the common difference, and the number of terms, you can find the sum of an arithmetic series using a simple formula.
The sum \( S_n \) of an arithmetic series can be calculated with the formula:

• \( S_n = \frac{n}{2} (a + l) \)

Where:

• \( n \) is the number of terms,
• \( a \) is the first term,
• \( l \) is the last term of the sequence.

Using this formula helps in quickly computing the sum without adding each term individually. This is especially useful for large sequences.

Sigma Notation

Sigma notation is a way to write sums in a compact form. The Greek letter \(\Sigma\) symbolizes 'sum' and is followed by an expression for the terms you are adding. It simplifies writing lengthy additions. Let's break down the parts of sigma notation.

• The expression inside the sigma (\( \Sigma \)) shows what you are summing, like \( 5i \)
• The variable below the \( \Sigma \), such as \( i=1 \), signals where you start the summation
• The number above the \( \Sigma \), like 6, indicates where you stop

Consider \( \sum_{i=1}^{6} 5i \) as an example:

• Start from \( i=1 \)
• Compute \( 5i \) for each \( i \) until \( i=6 \)
• Add all results to get the total

This approach allows mathematicians to efficiently express calculations for large or complex series.

Series Formulas

Series formulas provide a shortcut to finding the sum of sequences without having to manually add each term. They are essential in simplifying calculations, especially for long series.For arithmetic series, a common formula can be applied:

• The general formula is \( S_n = \frac{n}{2} (a + l) \) which we discussed earlier

When you have sums written with sigma notation, like \( \sum_{i=1}^{6} 5i \), you can use direct formulas to compute them faster. Steps often include finding the sum of natural numbers and then applying the multiplication factor (as in the original exercise).

• First, determine the sum of the numbers from 1 to \( n \)
• Use the formula \( \frac{n(n+1)}{2} \) for this sum
• If the expression involves a constant multiplier (e.g., \( 5 \) in \( 5i \)), multiply the sum by this constant

Such strategies not only save time but also improve accuracy, making them a crucial part of algebraic studies.