Problem 32 - A Graphical Approach to Precalculus with Limits

Question

Find the sum for each series. $$\sum_{i=1}^{6}(3 i-2)$$

Step-by-Step Solution

Answer

The sum of the series is 51.

Step 1: Identify the Series Formula

The given series is defined by \( \sum_{i=1}^{6}(3i-2) \). This means for each term in the series, we will use the formula \( 3i-2 \) where \( i \) is the index that goes from 1 to 6.

Step 2: Calculate Each Term in the Series

Calculate each term individually:- For \( i = 1 \), the term is \( 3(1)-2 = 1 \).- For \( i = 2 \), the term is \( 3(2)-2 = 4 \).- For \( i = 3 \), the term is \( 3(3)-2 = 7 \).- For \( i = 4 \), the term is \( 3(4)-2 = 10 \).- For \( i = 5 \), the term is \( 3(5)-2 = 13 \).- For \( i = 6 \), the term is \( 3(6)-2 = 16 \).

Step 3: Add the Terms Together

Add the terms calculated in Step 2:\( 1 + 4 + 7 + 10 + 13 + 16 = 51 \).

Key Concepts

Summation Notation

Summation notation is a neat and tidy way to express the addition of a sequence of terms. It is usually written with the Greek letter sigma (∑), followed by an expression that describes each term of the series. In the notation \( \sum_{i=1}^{6}(3i-2) \), the index \( i \) starts at 1 and goes up to 6.
Each time \( i \) increases, you substitute it into the expression \( 3i-2 \) to get a term in the series. Summation notation helps in simplifying the representation of long, repetitive additions and makes it easier to work with.
This way of writing saves space and clearly conveys the start and end points of the sequence.

• The expression inside the summation describes each term.
• The index \( i \) shows where to start and stop.
• The result is the total of all the terms from \( i = 1 \) to \( i = 6 \).

Series Formula

The series formula gives a rule for calculating each term of a sequence. In the example \( 3i-2 \), this formula provides a way to find the numbers you'll add together in the series.
Once the formula is known, you plug in each value of \( i \) from the range specified in the summation notation (e.g., from 1 to 6).
This allows you to compute each individual term.
The series formula is often used in the context of an arithmetic series where the difference between consecutive terms is constant.

• The formula \( 3i-2 \) is linear, meaning it increases at a steady rate.
• Each calculation is simple once \( i \) is substituted in.
• Understanding the formula helps anticipate the pattern of the series.

Arithmetic Sequence

An arithmetic sequence is a list of numbers with a common difference between consecutive terms. In every arithmetic sequence, the difference is constant, allowing easy prediction of future terms.
The given example, \( 3i-2 \), when calculated for \( i = 1 \) to \( i = 6 \), shows terms 1, 4, 7, 10, 13, and 16. Here, each term increases by 3, which is the common difference.
Arithmetic sequences can also be summed easily using specific formulas.

• The common difference helps in identifying the sequence type.
• Recognizing the pattern allows for simple calculations of further terms.
• Series formula targeting arithmetic sequences are very useful for quick summation.

Understanding arithmetic sequences aids in analyzing the progression of terms making the broader concept of series calculation intuitive.