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?
[tex]
\begin{array}{l}
\sum_{k=1}^{20}(2+6 k) \\
\sum_{k=1}^{20}(2+8 k)
\end{array}
[/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}=46[/tex]
How many stitches are in the first 20 rows of the shawl?
Answer (2)
Calculate the sum of the arithmetic series using the formula S n = 2 n ( a 1 + a n ) .
Substitute n = 20 , a 1 = 8 , and a 20 = 46 into the formula to find S 20 .
The sum of the first 20 terms is S 20 = 2 20 ( 8 + 46 ) = 540 .
The number of stitches in the first 20 rows of the shawl is 540 .
Explanation
Understanding the Problem We are given an arithmetic sequence representing the number of stitches in each row of a shawl. The first row has 8 stitches, and each subsequent row has 2 more stitches than the previous one. We are given the explicit formula for the number of stitches in the k th row: a k = 8 + ( k − 1 ) 2 . We want to find the total number of stitches in the first 20 rows, which is the sum S 20 . We also need to determine which of the two given summations is equal to S 20 .
Using the Arithmetic Series Sum Formula We can find the sum of the first 20 terms of the arithmetic series using the formula S n = 2 n ( a 1 + a n ) , where n is the number of terms, a 1 is the first term, and a n is the last term. In our case, n = 20 , a 1 = 8 , and a 20 = 46 .
Calculating the Sum Substituting the values into the formula, we get: S 20 = 2 20 ( 8 + 46 ) = 10 ( 54 ) = 540 So, there are 540 stitches in the first 20 rows of the shawl.
Evaluating the Summations Now, let's evaluate the two given summations:
∑ k = 1 20 ( 2 + 6 k ) = ∑ k = 1 20 2 + 6 ∑ k = 1 20 k = 2 ( 20 ) + 6 ⋅ 2 20 ( 20 + 1 ) = 40 + 6 ⋅ 2 20 ( 21 ) = 40 + 6 ⋅ 210 = 40 + 1260 = 1300
∑ k = 1 20 ( 2 + 8 k ) = ∑ k = 1 20 2 + 8 ∑ k = 1 20 k = 2 ( 20 ) + 8 ⋅ 2 20 ( 20 + 1 ) = 40 + 8 ⋅ 2 20 ( 21 ) = 40 + 8 ⋅ 210 = 40 + 1680 = 1720
Finding the Correct Summation Comparing the calculated value of S 20 with the values of the two summations, we see that neither of them is equal to S 20 = 540 . However, we were asked to find the number of stitches in the first 20 rows, which we found to be 540.
Final Answer The number of stitches in the first 20 rows of the shawl is 540.
Examples
Arithmetic series are useful in many real-life situations, such as calculating the total cost of items with increasing prices, determining the number of seats in an auditorium with each row having more seats than the previous one, or predicting the distance traveled by an object with increasing speed. In this case, we used an arithmetic series to calculate the total number of stitches in a knitted shawl, where each row has a fixed number of additional stitches compared to the previous row. Understanding arithmetic series helps in managing and predicting outcomes in scenarios involving sequential increments or decrements.
The total number of stitches in the first 20 rows of the shawl is 540. This is found using the formula for the sum of an arithmetic series, where the first term is 8 and the 20th term is 46. Hence, S 20 = 540 .
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