Problem 66 - A Graphical Approach to Precalculus with Limits

Question

Find the sum for each series. $$\sum_{i=1}^{5}(8 i-1)$$

Step-by-Step Solution

Answer

The sum of the series is 115.

Step 1: Understanding the Problem

We need to find the sum of the series represented by the expression \( \sum_{i=1}^{5}(8i-1) \). This means we will substitute the values from 1 to 5 into the expression \(8i-1\) and then find the sum of the results.

Step 1: Substitute Values into the Expression

Substitute \(i = 1\), \(i = 2\), \(i = 3\), \(i = 4\), and \(i = 5\) into the expression \(8i - 1\):- For \(i = 1\): \(8(1) - 1 = 8 - 1 = 7\)- For \(i = 2\): \(8(2) - 1 = 16 - 1 = 15\)- For \(i = 3\): \(8(3) - 1 = 24 - 1 = 23\)- For \(i = 4\): \(8(4) - 1 = 32 - 1 = 31\)- For \(i = 5\): \(8(5) - 1 = 40 - 1 = 39\)

Step 2: Calculate the Sum of the Series

Add all the calculated terms together:\[7 + 15 + 23 + 31 + 39\]

Step 3: Perform the Addition

Add the numbers step by step:1. \(7 + 15 = 22\)2. \(22 + 23 = 45\)3. \(45 + 31 = 76\)4. \(76 + 39 = 115\)

Step 5: Conclusion

The sum of the series \(\sum_{i=1}^{5}(8i-1)\) is 115.

Key Concepts

Summation Notation

When you see the symbol \(\sum\), think of it as a special way to indicate that we're adding up a series of numbers. It comes from the Greek letter sigma, which is a neat way to tell us to take a sum. A summation notation is useful because it gives a compact way to represent long expressions.

• The index variable, often \(i\), tells you where to start and stop adding. In our problem, \(i\) starts at 1 and ends at 5.
• The expression following the sigma tells us which numbers to add. Here, each term to add is given by \(8i - 1\).

Using summation notation can help organize complicated arithmetic tasks with many terms by outlining a clear process for finding the total. It allows us to communicate the same formula to find each term within the series, helping to maintain focus on simplifying and solving it.

Finite Series

A finite series represents the sum of a certain number of terms. The key word here is 'finite', meaning the series ends after a specific number of terms. In the example given, the series is finite because we are only adding five numbers.

• To be part of a finite series, each term has its own defined formula—in our problem, it’s \(8i - 1\).
• By knowing where to start and stop, we effectively find all the terms part of this sum, leading us to a complete total.

Finite series are found everywhere, from simple arithmetic progressions to complex financial computations. Understanding them involves breaking the problem into manageable parts, much like our formula \(8i - 1\), applying it to each index, and then summing the results. This clear structure helps in both learning and practical applications.

Step-by-Step Calculation

Calculating a series step-by-step is about dealing with one problem at a time. In our problem, this approach begins with evaluating each term of the series separately. First, substitute each value from 1 to 5 into the formula \(8i - 1\) according to the given steps:

• For \(i = 1\), compute \(8(1) - 1 = 7\).
• For \(i = 2\), compute \(8(2) - 1 = 15\).
• Continue similarly with the rest until \(i = 5\).

Once all terms are identified, the next step is straightforward addition of these terms. This is best done systematically:

• Start with the first two terms: add 7 and 15 to get 22.
• Then, add the next term to your running total: add 23 to get 45, and so forth.

Finally, you reach the sum of 115. Taking each step carefully ensures accuracy, and troubleshooting is easier since any mistake can be isolated to a specific part of the calculation.