Problem 53
Question
express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. $$ 1+3+5+\dots+(2 n-1) $$
Step-by-Step Solution
Verified Answer
The sum 1+3+5+...+(2n-1) can be represented in summation notation as \(\sum_{i=1}^{n}(2i - 1)\).
1Step 1: Understanding the series
This series is a sequence of odd numbers, which is an arithmetic sequence. Each term is an increment of two from the preceding term, and the first term is 1. Therefore, it can be said that the nth term is presented as 2n-1, where 'n' is the term number.
2Step 2: Representing in summation notation
The lower limit of the summation is '1' as given in the problem. Use 'i' as the index of the summation. The nth term of the sequence was identified as (2n-1) in Step 1, replace 'n' with 'i' as you are using 'i' as the index of the summation. Therefore, the given sum can be represented in summation notation as \(\sum_{i=1}^{n}(2i - 1)\).
Key Concepts
Arithmetic SequenceOdd NumbersIndex of SummationLower Limit of Summation
Arithmetic Sequence
An arithmetic sequence is a series of numbers such that the difference between consecutive terms is constant. This difference is known as the common difference. In the series given in the exercise, which starts at 1 and continues with odd numbers, we observe an arithmetic sequence with a common difference of 2.
Let’s break it down with the given example:
\(a_n = a_1 + (n-1)d\),
where \(a_1\) is the first term and \(d\) is the common difference.
For our sequence, the common difference \(d = 2\), and you can easily confirm that the nth term formula here is \(2n-1\). This shows that the series is perfectly aligned with the rules of an arithmetic sequence.
Let’s break it down with the given example:
- The first term is 1.
- The second term is 3, which is 1 plus 2.
- The third term is 5, which is 3 plus 2.
- And so on.
\(a_n = a_1 + (n-1)d\),
where \(a_1\) is the first term and \(d\) is the common difference.
For our sequence, the common difference \(d = 2\), and you can easily confirm that the nth term formula here is \(2n-1\). This shows that the series is perfectly aligned with the rules of an arithmetic sequence.
Odd Numbers
Odd numbers are integers not divisible by 2. They have the characteristic of being one more than the even number before them. In the sequence developed in the problem, all numbers are odd: 1, 3, 5, and so on.
This sequence can be described with the expression \(2n - 1\). Each term in this sequence adds 2 to the previous term because odd numbers increase incrementally by 2.
Therefore, when constructing a sequence or identifying odd numbers in a given series, remember the simple rule: start with 1 and continuously add 2 to reach the next term.
Recognizing this pattern helps in understanding the nature of numerical sequences involving odd numbers, especially when expressed in different mathematical forms such as series and sequences.
This sequence can be described with the expression \(2n - 1\). Each term in this sequence adds 2 to the previous term because odd numbers increase incrementally by 2.
Therefore, when constructing a sequence or identifying odd numbers in a given series, remember the simple rule: start with 1 and continuously add 2 to reach the next term.
Recognizing this pattern helps in understanding the nature of numerical sequences involving odd numbers, especially when expressed in different mathematical forms such as series and sequences.
Index of Summation
In summation notation, the index of summation is a variable, commonly denoted by \(i\), \(j\), \(k\), or other letters. It changes values starting from the specified lower limit to an upper limit, indicating which terms to sum.
In the exercise’s context, the index of summation is \(i\). It indicates that each term in the sequence will substitute \(i\) incrementally, starting from 1 up until \(n\).
In the exercise’s context, the index of summation is \(i\). It indicates that each term in the sequence will substitute \(i\) incrementally, starting from 1 up until \(n\).
- For example, when \(i = 1\), \(2i - 1 = 1\).
- For \(i = 2\), \(2i - 1 = 3\).
- And for \(i = n\), \(2n - 1\).
Lower Limit of Summation
The lower limit of summation is where the indexing begins in summation notation, indicating the first term to include in the sum. It is crucial as it sets the starting point for summation.
In our exercise, the lower limit of summation is 1. This means the summation will start with the value \(i=1\). The notation \(\sum_{i=1}^{n}\) specifies that we begin our additions with the first term and progress through the sequence.
Without specifying this starting point, the summation cannot be accurately computed, as it lacks a defined start and a particular sequence of terms to follow. By using a specific lower limit of summation, like 1 in this case, we ensure that each term from 1 up to \(n\) is included systematically.
In our exercise, the lower limit of summation is 1. This means the summation will start with the value \(i=1\). The notation \(\sum_{i=1}^{n}\) specifies that we begin our additions with the first term and progress through the sequence.
Without specifying this starting point, the summation cannot be accurately computed, as it lacks a defined start and a particular sequence of terms to follow. By using a specific lower limit of summation, like 1 in this case, we ensure that each term from 1 up to \(n\) is included systematically.
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