The given problem asks for the summation of \(\frac{m+1}{m}\) from \(n=1\) to \(n=3\). The summation notation \(\sum_{n=1}^{3}\) tells us to add the expression \(\frac{m+1}{m}\) for each value of \(n\) starting from 1 up to 3.
Notice that \(\frac{m+1}{m} = 1 + \frac{1}{m}\). This simplification breaks the expression into a simple sum of constants and fractions.
03
Write the Summation in a More Manageable Form
Since \(\frac{m+1}{m} = 1 + \frac{1}{m}\), we can rewrite the summation as: \(\sum_{n=1}^{3} \left( 1 + \frac{1}{m} \right) = \sum_{n=1}^{3} 1 + \sum_{n=1}^{3} \frac{1}{m}\).
04
Compute Each Part of the Summation
First, calculate \(\sum_{n=1}^{3} 1\). This is simply \(1 + 1 + 1 = 3\). Then calculate \(\sum_{n=1}^{3} \frac{1}{m}\). Since \(\frac{1}{m}\) does not depend on \(n\), this is equal to \(\frac{1}{m} + \frac{1}{m} + \frac{1}{m} = \frac{3}{m}\).
05
Combine the Results
Finally, add the results from the previous step: \(3 + \frac{3}{m} = 3 \left(1 + \frac{1}{m}\right)\).
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Series and Sequences
A sequence is an ordered list of numbers. Each number in a sequence is called a term. Sequences can be finite or infinite, depending on whether they have a last term. A series, on the other hand, is the sum of the terms of a sequence. Series can also be finite or infinite. For instance, when we are summing terms from 1 to 3 as in our example, we are working with a finite series. Understanding sequences and series is crucial in mathematics as they appear in various problems and applications. Noting the difference between them helps in solving problems correctly.
Algebraic Expressions
An algebraic expression is a combination of numbers, variables, and operators (such as + and -). In the given exercise, \(\frac{m+1}{m}\) is an algebraic expression. We simplified it to \(\frac{m+1}{m} = 1 + \frac{1}{m}\). Simplifying expressions is often useful as it can make calculations more straightforward. Each component of an algebraic expression can be individually handled and then combined for the final solution. Being comfortable with rewriting and simplifying algebraic expressions is essential for solving various mathematical problems efficiently. Practice breaking down complex expressions into simpler parts.
Finite Series
A finite series is a series that has a defined number of terms. The sum of the series in the exercise is a finite series since it sums from \ = 1\ to \ = 3\. To solve this, we broke the sum into manageable parts:
\( \sum_{n=1}^{3} 1 \)
\( \sum_{n=1}^{3} \frac{1}{m} \)
Adding these components gave us the final sum. When dealing with finite series, explicitly calculating each term can often lead to easier and clearer solutions. Remember to explore both the entire series and its individual parts to find the total sum efficiently.
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Most popular questions from this chapter
The expression \(a^{3}+3 a^{2} b+3 a b^{2}+b^{3}\) is called the expansion of\((a+b)^{3}\).A traffic light at an intersection has a 120 -sec cycle. The light is greenfor \(80 \mathrm{sec}\), yellow for \(5 \mathrm{sec}\), and red for \(35\mathrm{sec}\). a. When a motorist approaches the intersection, find the probability that thelight will be red. (Assume that the color of the light is defined as the colorwhen the car is \(100 \mathrm{ft}\) from the intersection. This is theapproximate distance at which the driver makes a decision to stop or go.) b. If a motorist approaches the intersection twice during the day, find theprobability that the light will be red both times.Determine whether the statement is true or false. If a statement is false,explain why. \(\sum_{i=1}^{n}\left(i^{2}-4 i+5\right)=\sum_{i=1}^{n} i^{2}-4 \sum_{i=1}^{n}i+5 n\)According to the Centers for Disease Control, the probability that a livebirth will be of twins in the United States is \(0.016\). What is theprobability that a live birth will not be of twins?Suppose that a box of DVDs contains 10 action movies and 5 comedies. a. If two DVDs are selected from the box with replacement, determine theprobability that both are comedies. b. It probably seems more reasonable that someone would select two differentDVDs from the box. That is, the first DVD would not be replaced before thesecond DVD is selected. In such a case, are the events of selecting comedieson the first and second picks independent events? c. If two DVDs are selected from the box without replacement, determine theprobability that both are comedies.
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We know that the sum of two numbers is the result obtained by adding two numbers. Thus, if {x1,x2,…,xn} { x 1 , x 2 , … , x n } is a sequence, then the sum of its terms is denoted using the symbol Σ (sigma). i.e., the sum of the above sequence = ∑ni=1xi=x1+x2+…. xn.
To find the sum of two or more numbers, you add them together. Start by writing down or mentally holding the numbers, then add each digit in the corresponding place value.
The symbol Σ (sigma) is generally used to denote a sum of multiple terms. This symbol is generally accompanied by an index that varies to encompass all terms that must be considered in the sum. For example, the sum of first whole numbers can be represented in the following manner: 1 2 3 ⋯.
Sum to n Terms of Arithmetic Progression Formula: Summing the first 'n' terms in an Arithmetic Progression (AP) is done with the formula: Sn = n/2 [2a + (n-1)d], where 'a' represents the initial term, 'd' is the consistent difference, and 'n' stands for the quantity of terms.
When we add two or more numbers, the result or the answer we get can be defined as the SUM. The numbers that are added are called addends. In the above example, 6 and 4 are addends, and 10 is their sum. In other words, we can say the sum of 8 and 5 is 13 or 8 added to 5 is 13.
The rule of sum is a basic counting approach in combinatorics. A basic statement of the rule is that if there are n choices for one action and m choices for another action, and the two actions cannot be done at the same time, then there are n + m n+m n+m ways to choose one of these actions.
A mathematical sum or maths sum is the result of adding two or more numbers together. It is the total of the numbers added together. For example, the sum of 3 and 7 is 10. They are taught to kids in their Maths lessons and can appear as numerical sums or can be structured as word problems.
The symbol ∈ indicates set membership and means “is an element of” so that the statement x∈A means that x is an element of the set A. In other words, x is one of the objects in the collection of (possibly many) objects in the set A.
The symbol Σ (capital sigma) is often used as shorthand notation to indicate the sum of a number of similar terms. Sigma notation is used extensively in statistics. For example, suppose we weigh five children. We will denote their weights by x1, x2, x3, x4 and x5.
The mathematical symbol "≠" represents "not equal to" or "is not equal to." It is used to indicate that two values or expressions are not the same. When you see "≠" between two entities, it means that they are distinct or different from each other. It is the negation of the equality symbol "=".
The sum of first n natural numbers as read above can be defined with the help of arithmetic progression. Where the sum of n terms is organized in a sequence with the first phase being with 1 and n being the number of terms along with the nth term. Sum of n numbers formula is [n(n+1)2].
the series in the last two cases being extended to n+1 terms. Prove that: 1 + 2 + 3 + ......... + n = n(n+1)2 i.e., the sum of the first n natural numbers is n(n+1)2.
Sum of n natural numbers can be defined as a form of arithmetic progression where the sum of n terms are arranged in a sequence with the first term being 1, n being the number of terms along with the nth term. The sum of n natural numbers is represented as [n(n+1)]/2.
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