What is the common difference $d$ of the arithmetic sequence on which the series is based?
$d=$ $\square$ -8
What is the explicit rule for the arithmetic sequence on which the series is based?
$a_k=\sqrt{ } 14+(k-1)(-8)$
What is the summation notation for the arithmetic series? Enter the correct values for $x, y$, and $z$.
$\begin{array}{l}<
\sum_{k=1}^x(y+z k) \\
x=\square \\
y=\square \\
z=\square\end{array}$
Answer (2)
Simplify the explicit rule: a k = 14 + ( k − 1 ) ( − 8 ) = 22 − 8 k .
Compare the general term with the summation notation form: a k = y + z k .
Identify the values: y = 22 and z = − 8 .
State the final values: x = x , y = 22 , z = − 8 , so the summation is ∑ k = 1 x ( 22 − 8 k ) .
Explanation
Understanding the Problem We are given an arithmetic sequence with a common difference d = − 8 and an explicit rule a k = 14 + ( k − 1 ) ( − 8 ) . We need to find the summation notation for the arithmetic series in the form ∑ k = 1 x ( y + z k ) , and determine the values of x , y , and z .
Simplifying the Explicit Rule First, let's simplify the explicit rule for the arithmetic sequence: a k = 14 + ( k − 1 ) ( − 8 ) = 14 − 8 k + 8 = 22 − 8 k So, the k -th term of the sequence is given by a k = 22 − 8 k .
Finding y and z Now, we want to express the arithmetic series in summation notation as ∑ k = 1 x ( y + z k ) . Comparing this with the general term a k = 22 − 8 k , we can identify the values of y and z . We have y = 22 and z = − 8 .
Writing the Summation Notation The summation notation for the arithmetic series is ∑ k = 1 x ( 22 − 8 k ) , where x represents the number of terms in the series. Since the number of terms is not specified, we leave x as a variable. Therefore, we have x = x , y = 22 , and z = − 8 .
Final Answer Thus, the values for the summation notation are: x = x y = 22 z = − 8
The summation notation is ∑ k = 1 x ( 22 − 8 k ) .
Examples
Arithmetic sequences and series are useful in many real-world scenarios. For example, consider a savings plan where you deposit a fixed amount each month. If you deposit $100 in the first month, and increase your deposit by $20 each subsequent month, the total amount you've saved after a certain number of months can be calculated using the sum of an arithmetic series. Understanding these concepts helps in financial planning and forecasting.
The common difference of the arithmetic sequence is d = − 8 . The explicit rule simplifies to a k = 22 − 8 k , identifying y = 22 and z = − 8 . The summation notation is s u m k = 1 x ( 22 − 8 k ) with x representing the number of terms.
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