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Magazine Chapter 14: Problem 45
2+8+14+20+\cdots+146 \quad 1850
Short Answer
Expert verified
The sum of the sequence is 1850.
Step by step solution
01
Identify the Pattern
The sequence given is: 2, 8, 14, 20, ..., 146. To solve this, we first need to recognize that it is an arithmetic sequence where each number increases by a common difference. Let's calculate the difference: 8 - 2 = 6, 14 - 8 = 6, 20 - 14 = 6. So, the common difference is 6.
02
Write the General Formula for the Sequence
An arithmetic sequence can be described by the formula \( a_n = a_1 + (n-1)d \), where \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) is the term number. Here, \( a_1 = 2 \) and \( d = 6 \). Thus, \( a_n = 2 + (n-1)6 \).
03
Determine the Number of Terms (n)
We are given that the last term is 146, so we set \( a_n \) equal to 146: \( 2 + (n-1)6 = 146 \).Solve for \( n \): \( (n-1)6 = 144 \)\( n-1 = 24 \)\( n = 25 \). Therefore, there are 25 terms in the sequence.
04
Calculate the Sum of the Sequence
The sum of an arithmetic sequence can be calculated using the formula: \( S_n = \frac{n}{2} (a_1 + a_n) \). Here, \( n = 25 \), \( a_1 = 2 \), and \( a_n = 146 \). Substitute these values in the formula: \( S_{25} = \frac{25}{2} (2 + 146) \).Calculate this: \( S_{25} = \frac{25}{2} \times 148 \)\( S_{25} = 25 \times 74 \)\( S_{25} = 1850 \).
05
Conclusion
The sum of the sequence 2 + 8 + 14 + 20 + ... + 146 is 1850, confirming the initial claim. Therefore, the calculations are correct and consistent with the sum provided.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Common Difference
In arithmetic sequences, the **common difference** is what defines the regular interval between consecutive terms. Recognizing this difference is key in identifying an arithmetic sequence.
Imagine you have a sequence: 2, 8, 14, 20, and so on. By calculating the difference between subsequent terms鈥�8 minus 2 gives us 6, then 14 minus 8 also gives 6鈥攜ou find that each number increases by 6.
Imagine you have a sequence: 2, 8, 14, 20, and so on. By calculating the difference between subsequent terms鈥�8 minus 2 gives us 6, then 14 minus 8 also gives 6鈥攜ou find that each number increases by 6.
- This consistent increase is what we call the common difference.
- In our example, the common difference is 6.
Sum of Arithmetic Sequence
Finding the **sum of an arithmetic sequence** is straightforward once you know the number of terms and the common difference. Using the formula for sum \[ S_n = \frac{n}{2} (a_1 + a_n) \]you can easily calculate it.
In this formula, \( S_n \) is the sum of the sequence, \( n \) is the number of terms, \( a_1 \) is the first term, and \( a_n \) is the last term.
It's a handy formula that saves us the trouble of adding each term manually, especially in larger sequences.
In this formula, \( S_n \) is the sum of the sequence, \( n \) is the number of terms, \( a_1 \) is the first term, and \( a_n \) is the last term.
- For our sequence (2, 8, 14, ..., 146), we have already found \( n = 25 \), \( a_1 = 2 \), and \( a_n = 146 \).
- Plug these into the formula: \( S_{25} = \frac{25}{2} (2+146) \).
It's a handy formula that saves us the trouble of adding each term manually, especially in larger sequences.
General Formula for Arithmetic Sequence
The **general formula for an arithmetic sequence** allows you to find any term within the sequence without having to list all the numbers leading up to it. This formula is \[ a_n = a_1 + (n-1)d \] where \( a_n \) is the term you wish to find, \( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference.
- In our example, the first term \( a_1 \) is 2, and the common difference \( d \) is 6.
- So, for any term \( a_n \), you plug these values into the formula: \( a_n = 2 + (n-1)6 \).
