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Magazine Chapter 10: Problem 32
Exer. 29-34: Express the sum in terms of summation notation. (Answers are not unique.) $$ 3+8+13+\cdots+463 $$
Short Answer
Expert verified
The sum is \(\sum_{n=1}^{93} (5n - 2)\).
Step by step solution
01
Identify the Sequence
The given sum is 3 + 8 + 13 + ... + 463. This represents an arithmetic sequence where each term after the first is obtained by adding a constant difference to the previous term.
02
Determine the Common Difference
To find the common difference (d), subtract the first term from the second term: \(d = 8 - 3 = 5\). So, the sequence has a common difference of 5.
03
Confirm the General Term
For an arithmetic sequence, the general term of the sequence can be expressed as \(a_n = a + (n-1) imes d\), where \(a\) is the first term, and \(d\) is the common difference. Here, \(a = 3\) and \(d = 5\), so the general term is \(a_n = 3 + (n-1) imes 5\).
04
Simplify the General Term
Simplifying the general term, we obtain: \(a_n = 3 + 5(n-1) = 5n - 2\). Thus, the \(n\)-th term of the sequence is \(5n - 2\).
05
Find the Number of Terms
To find how many terms are in the sequence, we set the \(n\)-th term equal to 463 and solve for \(n\): \(5n - 2 = 463\) \(5n = 465\) \(n = 93\) There are 93 terms in the sequence.
06
Express Sum in Summation Notation
Using the general term \(5n - 2\) and knowing there are 93 terms, we can express the total sum in summation notation as: \[\sum_{n=1}^{93} (5n - 2)\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Arithmetic Sequence
An arithmetic sequence is a series of numbers in which the difference between consecutive terms is constant. This constant difference is known as the "common difference." The sequence given in the problem, 3, 8, 13, ..., 463, is an arithmetic sequence. Here, each term after the first is formed by adding the common difference to the previous term.
Recognizing that an arithmetic sequence can always be represented as a linear pattern helps simplify many math problems. Each term gradually increases (or decreases) by this fixed step, making it predictable and hence, easier to sum up using summation notation.
Recognizing that an arithmetic sequence can always be represented as a linear pattern helps simplify many math problems. Each term gradually increases (or decreases) by this fixed step, making it predictable and hence, easier to sum up using summation notation.
Common Difference
The common difference in an arithmetic sequence is simply the amount by which we increase (or decrease for negative common differences) each time we move from one term to the next. In our sequence, the common difference is 5, calculated by subtracting the first term from the second term: 8 - 3 = 5.
This difference determines the "step" size of the sequence. Understanding the common difference is crucial because it influences the change between terms and thus plays a fundamental role in any computations involving the sequence, including finding terms or sums of terms.
This difference determines the "step" size of the sequence. Understanding the common difference is crucial because it influences the change between terms and thus plays a fundamental role in any computations involving the sequence, including finding terms or sums of terms.
General Term
The general term of an arithmetic sequence, sometimes referred to as the "nth term," represents any term in the sequence based on its position. It is expressed in the form of a formula: \( a_n = a + (n-1) \times d \) where:\( a \) is the first term, \( n \) is the term number, and \( d \) is the common difference.
For the given exercise, the sequence starts with 3, so \( a = 3 \) and \( d = 5 \). Therefore, the general term becomes:\( a_n = 3 + (n-1) \times 5 \).
Simplifying this gives \( a_n = 5n - 2 \), which helps in determining not only the value of any term but also the number of terms needed to sum to a particular value.
For the given exercise, the sequence starts with 3, so \( a = 3 \) and \( d = 5 \). Therefore, the general term becomes:\( a_n = 3 + (n-1) \times 5 \).
Simplifying this gives \( a_n = 5n - 2 \), which helps in determining not only the value of any term but also the number of terms needed to sum to a particular value.
Number of Terms
To determine how many terms are in a sequence, particularly when trying to sum it, it's crucial to figure out the number term (n) for the specific last value given. In our case, we need to find the term number for which the value is 463.
This is done by setting the general term equation equal to 463 and solving for \( n \):\( 5n - 2 = 463 \).First, solve \( 5n = 463 + 2 \), which gives \( 5n = 465 \). Dividing by 5 results in \( n = 93 \).
Thus, there are 93 terms in this arithmetic sequence, which helps conclude the sum using summation notation. Knowing the number of terms solidifies the understanding of the sequence's extent.
This is done by setting the general term equation equal to 463 and solving for \( n \):\( 5n - 2 = 463 \).First, solve \( 5n = 463 + 2 \), which gives \( 5n = 465 \). Dividing by 5 results in \( n = 93 \).
Thus, there are 93 terms in this arithmetic sequence, which helps conclude the sum using summation notation. Knowing the number of terms solidifies the understanding of the sequence's extent.
