A knitting pattern for a shawl calls for 8 stitches in the first row. Each row after that has 2 more stitches than the previous row. The explicit rule [tex]a_k=8+(k-1) 2[/tex] gives the number of stitches in the nth row.
Which summation is equal to [tex]S_{20}[/tex], the number of stitches in the first 20 rows of the shawl?
A. [tex]\sum_{k=1}^{20}(2+6 k)[/tex]
B. [tex]\sum_{k=1}^{20}(2+8 k)[/tex]
What is the first term of the arithmetic series?
[tex]a_1=8[/tex]
What is the 20th term of the arithmetic series?
[tex]a_{20}=\square[/tex]
Answer (2)
Calculate the 20th term using the explicit formula: a 20 = 8 + ( 20 − 1 ) 2 = 46 .
Calculate the sum of the first 20 terms using the arithmetic series formula: S 20 = 2 20 ( 8 + 46 ) = 540 .
Express the sum as a summation: S 20 = ∑ k = 1 20 ( 6 + 2 k ) .
Identify the correct summation from the given options: ∑ k = 1 20 ( 2 + 6 k ) . The 20th term is 46 .
Explanation
Problem Analysis We are given an arithmetic sequence where the first term is a 1 = 8 and the common difference is d = 2 . The explicit formula for the nth term is a k = 8 + ( k − 1 ) 2 . We need to find the 20th term, a 20 , and a summation that represents the sum of the first 20 terms, S 20 .
Finding the 20th Term First, let's find the 20th term, a 20 , using the given explicit formula: a 20 = 8 + ( 20 − 1 ) 2 = 8 + ( 19 ) 2 = 8 + 38 = 46 So, the 20th term is a 20 = 46 .
Calculating the Sum of the First 20 Terms Next, we need to find the sum of the first 20 terms, S 20 . We can use the formula for the sum of an arithmetic series: S n = 2 n ( a 1 + a n ) In our case, n = 20 , a 1 = 8 , and a 20 = 46 . Plugging these values into the formula, we get: S 20 = 2 20 ( 8 + 46 ) = 10 ( 54 ) = 540 So, the sum of the first 20 terms is S 20 = 540 .
Expressing the Sum as a Summation and Comparing with Options Now, let's express the sum S 20 as a summation. The general term of the sequence is a k = 8 + ( k − 1 ) 2 = 8 + 2 k − 2 = 6 + 2 k . Therefore, the sum of the first 20 terms can be written as: S 20 = k = 1 ∑ 20 a k = k = 1 ∑ 20 ( 6 + 2 k ) We can rewrite this summation as: S 20 = k = 1 ∑ 20 ( 2 k + 6 ) = k = 1 ∑ 20 ( 2 + 6 + 2 k − 2 ) = k = 1 ∑ 20 ( 2 + 2 ( 3 + k − 1 )) = k = 1 ∑ 20 ( 2 + 2 ( k + 2 )) = k = 1 ∑ 20 ( 2 + 2 k + 4 ) = k = 1 ∑ 20 ( 2 k + 6 ) Let's compare ∑ k = 1 20 ( 6 + 2 k ) with the given options. We can rewrite the summation as ∑ k = 1 20 ( 2 k + 6 ) .
The given options are: X ∑ k = 1 20 ( 2 + 6 k ) ∑ k = 1 20 ( 2 + 8 k )
Comparing ∑ k = 1 20 ( 2 k + 6 ) with the options, we see that the correct summation is ∑ k = 1 20 ( 2 + 6 k ) .
Final Answer Therefore, the 20th term of the arithmetic series is a 20 = 46 , and the summation that represents the sum of the first 20 terms is ∑ k = 1 20 ( 2 + 6 k ) .
Examples
Arithmetic sequences and series are useful in many real-life situations. For example, consider a savings plan where you deposit a fixed amount of money each month. If you deposit $100 in the first month and increase your deposit by $10 each subsequent month, the total amount you save over a certain period can be calculated using the sum of an arithmetic series. This helps you plan your savings and investments effectively. Another example is calculating the total cost of installing solar panels where the cost increases linearly with each additional panel.
The 20th term of the shawl's knitting pattern is a 20 = 46 , and the summation representing the total number of stitches in the first 20 rows corresponds to Option A: ∑ k = 1 20 ( 2 + 6 k ) .
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