Problem 31 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 29-34: Express the sum in terms of summation notation. (Answers are not unique.) $$ 4+11+18+\cdots+466 $$

Step-by-Step Solution

Answer

\(\sum_{n=1}^{67} (7n - 3)\)

Step 1: Identify the Sequence

Look for a pattern in the series provided. The series is given as: \(4, 11, 18, \ldots, 466\). Here, each term seems to increase by 7, indicating that the sequence is arithmetic with a common difference of 7.

Step 2: Define the General Term of the Arithmetic Sequence

The general form of an arithmetic sequence is given by \(a_n = a_1 + (n-1) \cdot d\), where \(a_1\) is the first term and \(d\) is the common difference. Substituting the given values, the general term \(a_n = 4 + (n-1)7 = 7n - 3\).

Step 3: Find the Number of Terms in the Sequence

We know the last term of the sequence is 466. Using the formula for the general term, set it equal to 466: \(7n - 3 = 466\). Solving for \(n\), we have \(7n = 469\), so \(n = 67\). There are 67 terms in the sequence.

Step 4: Express the Sum in Summation Notation

Using the general term \(7n - 3\) and the number of terms, express the sum using summation notation: \(\sum_{n=1}^{67} (7n - 3)\). This represents the sum of the arithmetic sequence from the first term to the 67th term.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a list of numbers where each term after the first is generated by adding a constant value to the previous term. This constant is known as the "common difference." In the sequence provided—4, 11, 18, ..., 466—we can see that each term increases by a consistent value, indicating that this sequence is arithmetic. Understanding arithmetic sequences helps in predicting the nth term and establishing a formula for the sequence.
By examining the difference between consecutive terms:

• 11 - 4 = 7
• 18 - 11 = 7

It is evident that the sequence increases by 7 each time. Therefore, to check if a sequence is arithmetic, look for this consistent difference between terms.

Common Difference

The common difference ( $d$) in an arithmetic sequence is crucial because it determines how the sequence progresses. In the sequence 4, 11, 18, ..., the common difference is 7, which we find by subtracting any term from the one that follows it. Defining the common difference allows us to construct a formula to find other terms in the sequence, making predictions easy and straightforward.

• If the common difference is positive, the sequence will grow.
• If it is negative, the sequence will decrease.
• Each term generated by this method contributes to forming the full arithmetic sequence.

Understanding this concept is essential to grasping how arithmetic sequences are formed and developed.

General Term

For arithmetic sequences, the general term formula is used to determine any term in the sequence without listing all preceding numbers. The formula is given by: \[ a_n = a_1 + (n-1) \cdot d \] where \( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference. In our example with the sequence 4, 11, 18, ..., 466, we substitute \( a_1 = 4 \) and \( d = 7 \) into the equation, resulting in \( a_n = 7n - 3 \).
This formula helps generate any term in the sequence by simply plugging the desired term's position (\( n \)) into the expression. It's a concise way to express the value of any term in the sequence without manual arithmetic every time.

Series

A series is the sum of terms in a sequence. For arithmetic sequences, the series can be expressed using summation notation, which gives a compact form of adding all terms from the first to the last. In summation notation, it is expressed as: \[ \sum_{n=1}^{N} (a_1 + (n-1) \cdot d) \] For the sequence in question, the sum is: \[ \sum_{n=1}^{67} (7n - 3) \] This notation clearly communicates that you want to sum the terms starting from \( n = 1 \) to the \( 67^{th} \) term, using the general term formula. Using summation notation is powerful because it provides a clear and systematic way of representing series, which is particularly useful in advanced mathematical studies.