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Chapter 7: Problem 36

Write the series using summation notation (starting with \(k=1\) ). Each series is either an arithmetic series or a geometric series. $$ 1+3+5+\cdots+201 $$

Short Answer

Expert verified
The summation notation for the given series is: \[\sum_{k = 1}^{101} (1 + 2(k - 1))\]

Step by step solution

01

Identify the arithmetic series

The given series is an arithmetic series with a common difference of 2 between consecutive terms: \(1, 3, 5, ...\)
02

Determine the general formula for the arithmetic series

The general form of an arithmetic series is given by: \(a_n = a_1 + (n - 1)d\) Where, \(a_n\) - The nth term of the series \(a_1\) - The first term of the series (in our case, \(a_1 = 1\)) \(n\) - The position of the term \(d\) - The common difference between consecutive terms (in our case, \(d = 2\)) So, the general formula for our series is: \(a_n = 1 + (n-1)2\)
03

Express the series using summation notation

Now we represent the series using summation notation, with \(k\) starting from 1: \[\sum_{k = 1}^n (1 + 2(k - 1))\]
04

Find the maximum value of \(k\)

We know that the last term of the series is 201, so setting \(a_n\) equal to 201, we can solve for \(n\): \(1 + (n-1)2 = 201\) Now, solve for \(n\): \(2(n - 1) = 200\) \((n - 1) = 100\) \(n = 101\)
05

Write the final summation notation

So, the summation notation for the given series is: \[\sum_{k = 1}^{101} (1 + 2(k - 1))\]

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Most popular questions from this chapter

Evaluate the arithmetic series. $$ 1001+1002+1003+\cdots+2998+2999+3000 $$

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