Evaluate the Summation sum from i=1 to 10 of 5+1/(2i)
The problem asks for the evaluation of a finite mathematical summation. Specifically, the summation starts with an index i=1 and ends with i=10. For each i value within this range, you are to calculate the term '5 + 1/(2i)' and then sum all of those terms together. The relation 1/(2i) suggests a series where each term includes the reciprocal of two times the index variable i, with 5 added to it. The objective is to perform this operation for each integer value of i from 1 to 10 and then add all those terms to find the total sum.
$\sum_{i = 1}^{10} 5 + \frac{1}{2 i}$
Answer
Solution:
Step 1
Decompose the given summation into two separate summations that are easier to handle. The summation of a sum is equal to the sum of the summations.
$$\sum_{i = 1}^{10} (5 + \frac{1}{2i}) = \sum_{i = 1}^{10} 5 + \sum_{i = 1}^{10} \frac{1}{2i}$$
Step 2
Calculate the summation of the constant term 5 over the range from 1 to 10.
Step 2.1
Use the summation formula for a constant:
$$\sum_{i = 1}^{n} c = cn$$
Step 2.2
Plug in the values for the constant summation:
$$5 \cdot 10$$
Step 2.3
Perform the multiplication to find the sum:
$$5 \cdot 10 = 50$$
Step 3
Now, evaluate the summation of the reciprocal of twice the index from 1 to 10.
Step 3.1
Write out the terms of the summation explicitly:
$$\frac{1}{2 \cdot 1} + \frac{1}{2 \cdot 2} + \frac{1}{2 \cdot 3} + \ldots + \frac{1}{2 \cdot 10}$$
Step 3.2
Calculate the sum of the terms:
$$\frac{7381}{5040}$$
Step 4
Combine the results of the two separate summations:
$$50 + \frac{7381}{5040}$$
Step 5
Simplify the combined result.
Step 5.1
Convert 50 to a fraction with the same denominator as the second term:
$$50 \cdot \frac{5040}{5040} + \frac{7381}{5040}$$
Step 5.2
Merge the two fractions by adding the numerators:
$$\frac{50 \cdot 5040}{5040} + \frac{7381}{5040}$$
Step 5.3
Add the numerators together over the common denominator:
$$\frac{50 \cdot 5040 + 7381}{5040}$$
Step 5.4
Finalize the simplification of the numerator.
Step 5.4.1
Multiply 50 by 5040:
$$\frac{252000 + 7381}{5040}$$
Step 5.4.2
Add the two numbers in the numerator:
$$\frac{259381}{5040}$$
Step 6
Express the result in various formats.
Exact Form:
$$\frac{259381}{5040}$$
Decimal Form:
$$51.46448412 \ldots$$
Mixed Number Form:
$$51 \frac{2341}{5040}$$
Knowledge Notes:
The problem involves evaluating a summation with a constant term and a variable reciprocal term. The key knowledge points and concepts used in solving this problem include:
Summation of a Constant: The sum of a constant $c$ over $n$ terms is simply $c$ multiplied by $n$:
$$\sum_{i = 1}^{n} c = cn$$
Separation of Summation Terms: A summation of a sum can be separated into the sum of individual summations:
$$\sum_{i = 1}^{n} (a_i + b_i) = \sum_{i = 1}^{n} a_i + \sum_{i = 1}^{n} b_i$$
Simplifying Fractions: To combine fractions, they must have a common denominator. Once they do, you can add the numerators and keep the common denominator.
Multiple Representations of Numbers: A number can be represented in various forms, such as an exact fraction, a decimal, or a mixed number.
Arithmetic Operations: Basic arithmetic operations such as multiplication and addition are used to simplify expressions.
Fraction Arithmetic: When adding fractions with different denominators, it is necessary to find a common denominator before combining them.
By understanding and applying these concepts, the problem can be solved step by step, leading to the final solution in various forms.
