Problem 49
Question
For Exercises \(45-50,\) write out the first three terms and the last term. Then use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum. $$\sum_{i=1}^{10} 4 i$$
Step-by-Step Solution
Verified Answer
The first three terms are 4, 8, 12 and the last term is 40. The sum of the first 10 terms is 220.
1Step 1: Identify the first three terms and the last term
The first term of the sequence is found when \(i = 1\). Thus, \(a_1 = 4 * 1 = 4\). \nThe second term is found when \(i = 2\). Thus, \(a_2 = 4 * 2 = 8\). \nThe third term is found when \(i = 3\). Thus, \(a_3 = 4 * 3 = 12\). \nThe tenth and last term is found when \(i = 10\). Thus, \(a_10 = 4 * 10 = 40\).
2Step 2: Apply the formula for the sum of first n terms
The formula for the sum of the first \(n\) terms of an arithmetic sequence is \(S_n = \frac{n}{2}(a_1 + a_n)\). Therefore, the sum \(S_{10}\) of the first 10 terms is given by \(S_{10} = \frac{10}{2}(a_1 + a_{10}) = 5 * (4 + 40) = 5 * 44 = 220\).
Key Concepts
Arithmetic SequenceSequence and SeriesAlgebraic Expressions
Arithmetic Sequence
An arithmetic sequence is a list of numbers with a constant difference between consecutive terms. To put it simply, if you start at a number and keep adding or subtracting the same value, you're creating an arithmetic sequence.
For example, if we start at 4 and add 4 each time, we get the sequence 4, 8, 12, 16, and so forth. This constant number that we add is called the 'common difference', denoted by 'd'. In our exercise, the first term, denoted by 'a_1', is 4 (when we plug in 'i = 1' into our formula), and our common difference is also 4 (since we're adding 4*i each time).
Understanding the pattern in these sequences is crucial because it allows us to predict future terms and even calculate the sum of a certain number of terms efficiently, without having to add each term individually. This capability is particularly useful in problems involving large sequences, where enumerating each term is impractical.
For example, if we start at 4 and add 4 each time, we get the sequence 4, 8, 12, 16, and so forth. This constant number that we add is called the 'common difference', denoted by 'd'. In our exercise, the first term, denoted by 'a_1', is 4 (when we plug in 'i = 1' into our formula), and our common difference is also 4 (since we're adding 4*i each time).
Understanding the pattern in these sequences is crucial because it allows us to predict future terms and even calculate the sum of a certain number of terms efficiently, without having to add each term individually. This capability is particularly useful in problems involving large sequences, where enumerating each term is impractical.
Sequence and Series
The terms 'sequence' and 'series' are often used interchangeably, but they have distinct meanings in mathematics. A sequence is an ordered list of numbers, whereas a series is the sum of the terms of a sequence.
In the context of arithmetic sequences, we often want to find the sum of a certain number of terms, which is where the concept of a series comes into play. Such series can be finite, like the one in our exercise involving the first 10 terms, or infinite if the number of terms grows without bound.
To calculate the sum of an arithmetic series, we use a special formula that leverages the structure of arithmetic sequences. This formula allows for quick and efficient calculations, bypassing the need for lengthy addition of each term. The understanding of series is instrumental in various fields, including finance, where it helps to determine the total value of investments or payments over time.
In the context of arithmetic sequences, we often want to find the sum of a certain number of terms, which is where the concept of a series comes into play. Such series can be finite, like the one in our exercise involving the first 10 terms, or infinite if the number of terms grows without bound.
To calculate the sum of an arithmetic series, we use a special formula that leverages the structure of arithmetic sequences. This formula allows for quick and efficient calculations, bypassing the need for lengthy addition of each term. The understanding of series is instrumental in various fields, including finance, where it helps to determine the total value of investments or payments over time.
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations that define a particular mathematical relationship. These expressions are the building blocks for formulating and solving equations.
In our exercise, the algebraic expression is given by \(4i\). It represents a family of numbers that one obtains by substituting different values of \(i\) into the expression. When \(i = 1, 2, 3,...,10\), we get the terms of the arithmetic sequence. The use of algebraic expressions provides a compact way to describe extensive sequences and allows us to generalize operations like finding the sum of sequences through formulas such as \(S_n = \frac{n}{2}(a_1 + a_n)\).
Algebraic expressions are indispensable for modeling real-world phenomena. By understanding how to manipulate and solve these expressions, we gain the ability to tackle a wide array of problems, from the simplest puzzles to the most complex scientific calculations.
In our exercise, the algebraic expression is given by \(4i\). It represents a family of numbers that one obtains by substituting different values of \(i\) into the expression. When \(i = 1, 2, 3,...,10\), we get the terms of the arithmetic sequence. The use of algebraic expressions provides a compact way to describe extensive sequences and allows us to generalize operations like finding the sum of sequences through formulas such as \(S_n = \frac{n}{2}(a_1 + a_n)\).
Algebraic expressions are indispensable for modeling real-world phenomena. By understanding how to manipulate and solve these expressions, we gain the ability to tackle a wide array of problems, from the simplest puzzles to the most complex scientific calculations.
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