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]\sum_{k=1}^{20}(2+6 k)[/tex]
[tex]\sum_{k=1}^{20}(2+8 k)[/tex]
What is the first term of the arithmetic series? [tex]a_1=[/tex] $\square$
Answer (2)
Find the first term: a 1 = 8 + ( 1 − 1 ) 2 = 8 .
Find the 20th term: a 20 = 8 + ( 20 − 1 ) 2 = 46 .
Calculate the sum of the first 20 terms: S 20 = 2 20 ( 8 + 46 ) = 540 .
Express the sum in summation notation: S 20 = ∑ k = 1 20 ( 2 k + 6 ) .
The first term is 8 .
Explanation
Understanding the Problem The problem provides an explicit rule for the number of stitches in the nth row of a knitting pattern: a k = 8 + ( k − 1 ) 2 . We need to find the summation that represents the total number of stitches in the first 20 rows, denoted as S 20 , and also determine the first term of the arithmetic series, a 1 .
Finding the First Term First, let's find the first term, a 1 , by substituting k = 1 into the explicit rule: a 1 = 8 + ( 1 − 1 ) 2 = 8 + 0 = 8
Finding the 20th Term Next, we need to find the number of stitches in the 20th row, a 20 , by substituting k = 20 into the explicit rule: a 20 = 8 + ( 20 − 1 ) 2 = 8 + ( 19 ) 2 = 8 + 38 = 46
Calculating the Sum of the First 20 Terms Now, we can find the sum of the first 20 terms, S 20 , using 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 . Therefore, S 20 = 2 20 ( 8 + 46 ) = 10 ( 54 ) = 540
Expressing the Sum in Summation Notation We need to express S 20 using summation notation. The general term of the arithmetic series is a k = 8 + ( k − 1 ) 2 = 8 + 2 k − 2 = 6 + 2 k = 2 k + 6 . Therefore, the sum of the first 20 terms can be written as: S 20 = k = 1 ∑ 20 ( 2 k + 6 ) = k = 1 ∑ 20 ( 6 + 2 k ) Comparing this with the given options, we see that it matches ∑ k = 1 20 ( 2 + 6 k ) is not correct, but ∑ k = 1 20 ( 6 + 2 k ) is equivalent to ∑ k = 1 20 ( 2 k + 6 ) .
Final Answer The summation equal to S 20 is ∑ k = 1 20 ( 2 k + 6 ) . The first term of the arithmetic series is a 1 = 8 .
Examples
Arithmetic series are useful in many real-life situations. For example, consider a stack of logs where each layer has one less log than the layer below it. If the bottom layer has 20 logs, and the top layer has 1 log, we can use an arithmetic series to calculate the total number of logs in the stack. Another example is calculating the total cost of buying an item over time with increasing payments. Understanding arithmetic series helps in managing finances and predicting outcomes in scenarios with consistent incremental changes.
The correct summation that equals S 20 is ∑ k = 1 20 ( 2 k + 6 ) . The first term of the series is a 1 = 8 .
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