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Chapter 10: Problem 22

In Problems \(17-22\), find the sum of the given arithmetic series. $$ -5-3-1+\cdots+25 $$

Short Answer

Expert verified
The sum of the series is 160.

Step by step solution

01

Identify the Sequence

The given series is an arithmetic sequence because the difference between consecutive terms is consistent. The first term \(a_1\) is \(-5\), and the sequence continues as \(-3, -1, \ldots, 25\).
02

Determine the Common Difference

The common difference \(d\) between terms can be found by subtracting the first term from the second term: \(-3 - (-5) = 2\). Thus, \(d = 2\).
03

Calculate the Number of Terms

The last term \(a_n\) of the sequence is 25. Use the formula for the \(n\)-th term of an arithmetic sequence: \(a_n = a_1 + (n-1)d\). Substitute the known values to find \(n\): \[25 = -5 + (n-1) \times 2.\]Solve for \(n\): \[25 = -5 + 2n - 2,\]\[25 = 2n - 7,\]\[32 = 2n,\]\[n = 16.\] Thus, there are 16 terms in this sequence.
04

Use the Sum Formula for an Arithmetic Series

The sum \(S_n\) of an arithmetic series is calculated with the formula: \[S_n = \frac{n}{2} (a_1 + a_n).\] Substitute the values for \(n = 16\), \(a_1 = -5\), and \(a_n = 25\):\[S_{16} = \frac{16}{2}(-5 + 25).\] Calculate the sum:\[S_{16} = 8 \times 20 = 160.\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Arithmetic Sequence
An arithmetic sequence is a list of numbers where each term after the first is obtained by adding a constant to the previous term. This constant is known as the common difference. In the sequence given, the terms are
  • -5
  • -3
  • -1
  • ... and so on, up to 25.
This shows that the sequence adds the same value each time to get from one number to the next. Essentially, understanding an arithmetic sequence allows you to predict any given term if you know the first term and the common difference.
Recognizing the pattern in arithmetic sequences helps not just in identifying the sequences but also in describing them mathematically.
Common Difference
The common difference in an arithmetic sequence is the amount you'd add (or subtract if negative) to move from one term to the next. To find it, subtract any two consecutive terms. For example, if you are given the sequence
  • -5, -3
We find the difference by calculating
  • -3 - (-5) = 2.

The common difference here is 2, indicating each term is just 2 more than the previous one. This common difference is crucial, as it provides the rule that generates the entire sequence.
Understanding the common difference can help you pinpoint any term in the sequence or even find the number of terms if given the last term and the first one.
Sum Formula
The sum formula for an arithmetic series allows you to find the total of all numbers in the sequence quickly and efficiently. If you're tasked with summing all terms of an arithmetic sequence, the formula you use is: \[S_n = \frac{n}{2} (a_1 + a_n)\]where:
  • \(S_n\) is the sum of the first \(n\) terms,
  • \(a_1\) is the first term,
  • \(a_n\) is the last term, and
  • \(n\) is the number of terms.
This formula ensures that you don鈥檛 have to add each term one by one. Applying it to the sequence
  • -5 to 25
gives you a total sum of 160, knowing there are 16 terms between -5 and 25. Using this formula, summing becomes a matter of plugging in the known values, making it both simple and reliable.

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