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Chapter 8: Problem 56

Express each sum using summation notation. Use a lower limit of summation of your choice and k for the index of summation. $$6+8+10+12+\dots+32$$

Short Answer

Expert verified
The series \(6+8+10+12+...+32\) can be written in summation notation as \(\sum_{k=1}^{14} (4 + 2k)\).

Step by step solution

01

Identify the Common Difference

The numbers in the series are increasing by 2, therefore the common difference, \(d\), is 2.
02

Identify the First Term

The first term in this arithmetic series, \(a_1\), is 6.
03

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 \(a_k = 6 + (k - 1)2 = 4 + 2k\).
04

Write in Summation Notation

The series can now be written in summation notation as \(\sum_{k=1}^{14} (4 + 2k)\)

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