Evaluate the Summation sum from n=1 to 5 of 1/(n+4)
The problem presented asks for the calculation of a finite mathematical series. Specifically, it is requesting to find the sum of the terms of the form 1/(n+4) as the variable "n" takes on integer values starting from 1 to 5. This involves computing the individual fractions 1/5, 1/6, 1/7, 1/8, and 1/9, and then adding them together to get the total sum.
$\sum_{n = 1}^{5} \frac{1}{n + 4}$
Answer
Solution:
Step 1:
Write out the terms of the summation for each integer value of $n$ from 1 to 5.
$$\frac{1}{1 + 4} + \frac{1}{2 + 4} + \frac{1}{3 + 4} + \frac{1}{4 + 4} + \frac{1}{5 + 4}$$
Step 2:
Calculate the sum of the terms.
$$\frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} + \frac{1}{9} = \frac{1879}{2520}$$
Step 3:
Present the final sum in its various representations.
- Exact Form: $\frac{1879}{2520}$
- Decimal Form: $0.74563492 \ldots$
Knowledge Notes:
The problem involves evaluating a finite summation, which is a process of adding up a sequence of numbers generated by a function of the summation index. The summation index here is $n$, and it runs from 1 to 5. The function inside the summation is $\frac{1}{n+4}$.
To solve this, we follow these steps:
Expansion: We expand the summation by substituting each value of $n$ into the function and writing out the terms.
Simplification: We simplify the expanded form by performing the arithmetic operations to find the sum.
Representation: The result can be expressed in different forms, such as an exact fraction or a decimal approximation.
Relevant knowledge points include:
Understanding of summation notation and how to expand it.
Basic arithmetic operations on fractions.
Conversion between exact fractions and decimal representations.
Simplification techniques for fractions, including finding a common denominator and adding numerators.
