Problem 38 - Beginning and Intermediate Algebra

Question

Explain what \(\sum_{i=1}^{n}(7 i+2)\) means.

Step-by-Step Solution

Answer

The expression \(\sum_{i=1}^{n}(7 i+2)\) represents the sum of elements calculated using the formula \(7 i + 2\), where \(i\) is the index that ranges from 1 to n. Each element in the sum is calculated for an integer value of \(i\), and all the elements are added together. In a practical context, this sum might represent a quantity that accumulates over time or follows a rule, such as a weekly bonus that increases according to the formula \(7i + 2\), where \(i\) represents the number of weeks passed. The summation would then represent the total bonus after n weeks.

Step 1: Summation notation

The given expression, \(\sum_{i=1}^{n}(7 i+2)\), represents a sum of elements, where each element is calculated using the formula inside the parentheses. The symbol \(\sum\) is called the summation symbol, and the variables below and above it indicate the range of the sum: \(i\) is the index that ranges from the starting value (1 in this case) to the ending value (n in this case).

Step 2: Elements of the sum

The formula inside the parentheses, \(7 i + 2\), indicates the value of each element we add in the sum. For each value of the index \(i\) from 1 to n, we calculate \(7 i + 2\) and add it to the sum.

Step 3: Calculating the sum

To calculate the sum \(\sum_{i=1}^{n}(7 i+2)\), we follow these steps: 1. Replace \(i\) with 1 in the formula \(7 i + 2\), which gives us \(7 (1) + 2 = 9\) 2. Replace \(i\) with 2 in the formula \(7 i + 2\), which gives us \(7 (2) + 2 = 16\) 3. Continue this process for all integer values of \(i\) from 1 to n 4. Add all the elements we computed, that is: \(9 + 16 + \dots + 7n + 2\)

Step 4: Practical context

In a practical context, this sum might represent a quantity that is accumulated over time or added together by some rule. For example, let's say we have a student who earns a bonus every week, and the bonus increases weekly according to the formula \(7i + 2\), where \(i\) is the number of weeks passed. This summation \(\sum_{i=1}^{n}(7 i+2)\) would represent the total bonus the student has after n weeks.

Key Concepts

Series and Sequences

In mathematics, a series is essentially the sum of the terms of a sequence. A sequence itself is an ordered list of numbers where each number in the list is called a term. For example, if you have a sequence of numbers derived from the formula \(7i + 2\), for values of \(i\) starting from 1 to \(n\), each term can be represented as \(9, 16, 23,\) and so on, depending on the values of \(i\).

When you take these numbers and add them together, it becomes a series. The expression \(\sum_{i=1}^{n}(7 i+2)\) is the notation that represents this series, indicating that we add up each term calculated by the formula \(7i + 2\) for values of \(i\) from 1 through \(n\).

This method can be visualized as stacking individual blocks one on top of another, where each block represents a calculated term, creating a cumulative total or series.

Indexing

Indexing in sequences and series is the method of identifying each element or term by an index number. The index, often denoted by \(i\), indicates the position of a term within the sequence.

In the summation notation \(\sum_{i=1}^{n}(7 i+2)\), \(i\) begins at 1 and increases incrementally by 1 until it reaches the value of \(n\). The numbers below and above the summation symbol \(\sum\) specify this range: the expression "\(i = 1\)" starts the index, while "\(n\)" tells us where to stop.

This process of indexing allows us to systematically calculate each term in the sequence, ensuring that we cover every position from start to finish. It acts like an address for each term, keeping our process organized and structured. This precise management of sequence setup means we efficiently handle calculations without missing any steps.

Sum Calculation

Calculating the sum in the expression \(\sum_{i=1}^{n}(7 i+2)\) involves a series of steps where you replace the index \(i\) with each integer in its range and compute each term.

Here's a breakdown:

• Start with \(i = 1\), substitute it in the formula to get the first term: \(7(1) + 2 = 9\).
• Then, increment \(i\) to 2, calculate the next term: \(7(2) + 2 = 16\).
• Continue this process until \(i = n\), each time calculating \(7i + 2\).
• Add together all the terms you have obtained: \(9 + 16 + \ldots + (7n + 2)\). This sum represents the result of the series.

Using this straightforward method ensures a complete and accurate sum calculation. By approaching it step-by-step, you simplify what might initially seem complex, making it easier to grasp the principles behind summation notation.