Problem 10
Question
Write the indicated sum in sigma notation. \(2+4+6+8+\cdots+50\)
Step-by-Step Solution
Verified Answer
\(\sum_{n=1}^{25} 2n\)
1Step 1: Identify the Pattern
Examine the sequence given in the sum: 2, 4, 6, 8, ..., 50. Notice that each term is an even number, and the sequence increases by 2 each time. This sequence is an arithmetic sequence with the first term 2 and a common difference of 2.
2Step 2: Determine the Number of Terms
To write the series in sigma notation, we need to know the total number of terms. We know the formula for the n-th term of an arithmetic sequence is \(a_n = a + (n-1) \,d\), where \(a\) is the first term and \(d\) is the common difference. Here, \(50 = 2 + (n-1) \,2\). Solve for \(n\): \(50 = 2 + 2n - 2\) simplifies to \(50 = 2n\), giving \(n = 25\). Thus, there are 25 terms.
3Step 3: Write in Sigma Notation
Now, write the sum using sigma notation. The general form for the terms is \(2n\) where \(n\) ranges from 1 to 25. Hence, the series can be expressed in sigma notation as \(\sum_{n=1}^{25} 2n\).
Key Concepts
Arithmetic SequenceSum of SequenceNumber of Terms
Arithmetic Sequence
An arithmetic sequence is a collection of numbers that follow a specific pattern. Each number in the sequence is called a term. The unique characteristic of an arithmetic sequence is that the difference between successive terms is constant. This difference is known as the common difference.
To determine if a sequence is arithmetic, examine the difference between consecutive terms. If this difference remains consistent throughout the sequence, it confirms the arithmetic nature. For the sequence 2, 4, 6, 8, ..., 50, the difference between each pair of consecutive terms is 2. This reveals it's an arithmetic sequence.
Another aspect of arithmetic sequences involves finding any term. The formula for the nth term is given by:
\[a_n = a + (n-1) \, d\]
where \(a\) is the first term and \(d\) is the common difference.
To determine if a sequence is arithmetic, examine the difference between consecutive terms. If this difference remains consistent throughout the sequence, it confirms the arithmetic nature. For the sequence 2, 4, 6, 8, ..., 50, the difference between each pair of consecutive terms is 2. This reveals it's an arithmetic sequence.
Another aspect of arithmetic sequences involves finding any term. The formula for the nth term is given by:
\[a_n = a + (n-1) \, d\]
where \(a\) is the first term and \(d\) is the common difference.
Sum of Sequence
The sum of an arithmetic sequence, also known as an arithmetic series, can quickly be determined using a formula. The formula for finding the sum of the first \(n\) terms of an arithmetic sequence is:
\[S_n = \frac{n}{2} \times (a + l)\]
In this context, \(S_n\) is the sum of the sequence, \(n\) is the number of terms, \(a\) is the first term, and \(l\) is the last term of the sequence.
Let’s apply this to the sequence 2, 4, 6, ..., 50 to find its sum. We have already determined that there are 25 terms.
\[S_{25} = \frac{25}{2} \times (2 + 50) = \frac{25}{2} \times 52 = 650\]
Thus, the total sum of this arithmetic sequence is 650.
\[S_n = \frac{n}{2} \times (a + l)\]
In this context, \(S_n\) is the sum of the sequence, \(n\) is the number of terms, \(a\) is the first term, and \(l\) is the last term of the sequence.
Let’s apply this to the sequence 2, 4, 6, ..., 50 to find its sum. We have already determined that there are 25 terms.
- First term (\(a\)) = 2
- Last term (\(l\)) = 50
- Number of terms (\(n\)) = 25
\[S_{25} = \frac{25}{2} \times (2 + 50) = \frac{25}{2} \times 52 = 650\]
Thus, the total sum of this arithmetic sequence is 650.
Number of Terms
Calculating the number of terms in an arithmetic sequence is a fundamental step when expressing a series in sigma notation. To find the number of terms, use the formula for the nth term:
\[a_n = a + (n-1) \, d\]
You can manipulate this formula to solve for \(n\) (number of terms). For example, consider the sequence 2, 4, 6, ..., 50, where:
\[50 = 2 + (n-1) \, 2\]
Simplifying, we get \(50 = 2n\), hence \(n = 25\).
Therefore, this sequence contains 25 terms. Knowing the number of terms helps to express the entire sequence concisely in sigma notation, making calculations easier and clearer.
\[a_n = a + (n-1) \, d\]
You can manipulate this formula to solve for \(n\) (number of terms). For example, consider the sequence 2, 4, 6, ..., 50, where:
- First term (\(a\)) = 2
- Last term (\(a_n\)) = 50
- Common difference (\(d\)) = 2
\[50 = 2 + (n-1) \, 2\]
Simplifying, we get \(50 = 2n\), hence \(n = 25\).
Therefore, this sequence contains 25 terms. Knowing the number of terms helps to express the entire sequence concisely in sigma notation, making calculations easier and clearer.
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