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Chapter 14: Problem 45

Write each series using summation notation. $$ 3+4+5+6+7 $$

Short Answer

Expert verified
Summation notation: \(\sum_{i=1}^{5}(3+(i-1) \times 1)\).

Step by step solution

01

Identify the Pattern

Observe the given series: 3, 4, 5, 6, 7. Notice that the series is an arithmetic sequence where the difference between consecutive terms is 1.
02

Determine the First Term and the Last Term

The first term (\(a\)) of the series is 3 and the last term (\(l\)) is 7.
03

Determine the Number of Terms

Calculate the number of terms (\(n\)): The formula for the nth term of an arithmetic sequence is \(a + (n-1)d\), where \(d\) is the common difference. Here, \(d = 1\). Solving for \(n\) gives \(3 + (n-1) \times 1 = 7\). Simplifying, we get \(3 + n - 1 = 7\), thus \(n = 5\).
04

Express in Summation Notation

The general form in summation notation for an arithmetic series is \(\sum_{i=1}^{n}(a + (i-1)d)\). Applying the values, the series can be written as \(\sum_{i=1}^{5}(3 + (i-1) \times 1)\).

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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 sequence of numbers in which the difference between any two consecutive terms is always the same. This difference is called the common difference. For example, the sequence 3, 4, 5, 6, 7 is an arithmetic sequence because the difference between each consecutive term is consistently 1. Arithmetic sequences can be found in various aspects of daily life, such as calendar dates and save table distances.
first term
The first term of an arithmetic sequence is simply the initial number in the sequence. It is usually denoted by the symbol \(a\). In our example, the sequence is 3, 4, 5, 6, 7, so the first term is 3. Knowing the first term is crucial when using formulas to further analyze or express the sequence. Whenever you're examining sequences, always start by identifying the first term.
number of terms
The number of terms in an arithmetic sequence is the total count of numbers present in the sequence. It is typically represented by the symbol \(n\). To find \(n\) when you know the first term \(a\), the common difference \(d\), and the last term \(l\), you can use the nth term formula. For our sequence 3, 4, 5, 6, 7, we know the common difference is 1 and the last term is 7. Solving the equation \(3 + (n-1) \times 1 = 7\) gives us \(n = 5\). Thus, there are 5 terms in this sequence.
common difference
The common difference in an arithmetic sequence is the consistent difference between consecutive terms. It is denoted by \(d\). You find it by subtracting any term from the subsequent term in the sequence. For example, in the sequence 3, 4, 5, 6, 7, the common difference \(d\) is 4 - 3 = 1. This value remains constant throughout the sequence and plays a key role in various calculations involving the sequence, such as finding any term or expressing the sequence using summation notation.

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