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Chapter 11: Problem 9

Match the expression with the most appropriate expression from the column on the right. ____ \(5+10+15+20\) a) \(-1+1+(-1)+1\) b) \(a_{2}=25\) c) \(a_{2}=8\) d) \(\sum_{k=1}^{4} 5 k\) e) \(S_{3}\) f) \(1+4+9+16\)

Short Answer

Expert verified
d) \(\sum_{k=1}^{4} 5k\)

Step by step solution

01

Analyze the given expression

The given expression is \(5+10+15+20\). Observe the pattern or series type in this expression.
02

Identify the pattern in the given expression

The expression \(5+10+15+20\) represents the sum of multiples of 5: \(5 \times 1 + 5 \times 2 + 5 \times 3 + 5 \times 4\).
03

Translate the given expression to summation form

Notice that this can be written as \( \sum_{k=1}^{4} 5k\)\.
04

Match with the options

Review the options to find which best represents the summation form \(\sum_{k=1}^{4} 5k\).The correct option is d) \(\sum_{k=1}^{4} 5k\).

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Key Concepts

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

series
In mathematics, a series is the sum of the terms of a sequence. Sequences are ordered lists of numbers, and when you add these numbers together, you get a series. These can be finite or infinite. In a finite series, the sum covers a set number of terms. For example, in the expression \( 5 + 10 + 15 + 20 \), we only add four numbers together, making it a finite series. Some common types of series include arithmetic and geometric series. The given exercise deals with an arithmetic series.
pattern recognition
Pattern recognition in mathematics involves identifying a repeating or regular sequence. Recognizing patterns helps simplify complex problems and convert expressions into simpler, more manageable forms. Let's take the example \(5 + 10 + 15 + 20\).
You might see a pattern where each number increases by 5. This consistent addition indicates a pattern. By recognizing that each term is 5 more than the previous term, you can identify that this is an arithmetic pattern. This pattern recognition is a crucial step in converting the sequence into a summation expression, such as \(\sum \limits _{k=1}^{4} 5k\).
arithmetic sequence
An arithmetic sequence is a list of numbers with a common difference between consecutive terms. For example, in the sequence \(5, 10, 15, 20\), you add 5 each time to get the next term. The general form of an arithmetic sequence can be written as:
\ a, a+d, a+2d, a+3d,...
Here, \(a\) is the first term, and \(d\) is the common difference. In our earlier expression \(5 + 10 + 15 + 20\), \(a = 5\) and \(d = 5\). Recognizing an arithmetic sequence helps in converting these expressions into summation notation, making it easier to analyze and solve specific problems.

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Most popular questions from this chapter

use the formula for \(S_{n}\) to find the indicated sum for each geometric series. $$S_{6} \text { for } 16-8+4-\cdots$$

Perform the indicated operation and, if possible, simplify. $$ \frac{t}{t-1}-\frac{t}{1-t} $$

Find the first five terms of each sequence; then find \(S_{5}\). $$ a_{n}=\frac{1}{2^{n}} \log 1000^{n} $$

Write out and evaluate each sum. $$ \sum_{k=2}^{5} \frac{k-1}{k+1} $$

Prove that if \(p, m,\) and \(q\) are consecutive terms in an arithmetic sequence, then $$ m=\frac{p+q}{2} $$
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