Problem 56
Question
Express each sum using summation notation. Use a lower limit of summation of your choice and k for the index of summation. $$6+8+10+12+\dots+32$$
Step-by-Step Solution
Verified Answer
The series \(6+8+10+12+...+32\) can be written in summation notation as \(\sum_{k=1}^{14} (4 + 2k)\).
1Step 1: Identify the Common Difference
The numbers in the series are increasing by 2, therefore the common difference, \(d\), is 2.
2Step 2: Identify the First Term
The first term in this arithmetic series, \(a_1\), is 6.
3Step 3: Determine the General Term
The general form of an arithmetic sequence is \(a_k = a_1 + (k - 1)d\), where \(a_k\) is the value of the term at position \(k\), \(a_1\) is the first term, \(d\) is the common difference, and \(k\) is the position. In this case, the arithmetic sequence is \(a_k = 6 + (k - 1)2 = 4 + 2k\).
4Step 4: Write in Summation Notation
The series can now be written in summation notation as \(\sum_{k=1}^{14} (4 + 2k)\)
Other exercises in this chapter
Problem 56
Write a probability word problem whose answer is one of the following fractions: \(\frac{1}{6}\) or \(\frac{1}{4}\) or \(\frac{1}{3}\)
View solution Problem 56
Find the middle term in the expansion of $$\left(\frac{1}{x}-x^{2}\right)^{12}$$.
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Solve by the method of your choice. In a race in which six automobiles are entered and there are no ties, in how many ways can the first four finishers come in?
View solution Problem 57
Explain how to find the probability of an event not occurring. Give an example.
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