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Chapter 7: Problem 35

Write the series using summation notation (starting with \(k=1\) ). Each series is either an arithmetic series or a geometric series. $$ 2+4+6+\cdots+100 $$

Short Answer

Expert verified
The given series in summation notation is: \[ \sum_{k=1}^{50} (2 + 2(k-1)) \]

Step by step solution

01

Identify the Type of Series and the General Term

In order to determine if the series is arithmetic or geometric, we need to check if the difference between consecutive terms is constant (arithmetic) or if the ratio between consecutive terms is constant (geometric). By comparing consecutive terms: \(4 - 2 = 6 - 4 = 2\) The difference between consecutive terms is a constant, which is 2. Therefore, this series is an arithmetic series. The general term for an arithmetic sequence can be written as: \( a_n = a_1 + (n - 1)d \) In this case, the first term, a1, is 2, and the common difference, d, is also 2.
02

Identify the Number of Terms (n)

We know the last term of this series is 100, and we also know the general term formula for an arithmetic sequence, which is \( a_n = a_1 + (n - 1)d \). We can use this to find the number of terms in the sequence. \(100 = 2 +(n - 1)2\) Now we need to solve the equation for n.
03

Solve for n

Given the equation: \(100 = 2 +(n - 1)2\) Step 1: Subtract 2 from both sides of the equation. \(98 = (n - 1)2\) Step 2: Divide both sides of the equation by 2. \(49 = n - 1\) Step 3: Add 1 to both sides of the equation. \(50 = n\) So, there are a total of 50 terms in the sequence.
04

Write the Summation Notation

Now that we have all the necessary information (arithemtic series with general term formula, \(a_n = 2 + 2(n - 1)\) and n = 50), we can write the series using summation notation. The formula for the arithmetic series using summation notation is: \( \sum_{k=1}^{n} a_k = a_1+(k-1)d \) Which can be written for our series as: \( \sum_{k=1}^{50} (2 + 2(k-1)) \) Therefore, the given series in summation notation is: \[ \sum_{k=1}^{50} (2 + 2(k-1)) \]

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