Evaluate the Summation sum from n=2 to 5 of 4n
The problem asks for the calculation of a finite series where the term being summed is four times the variable n (4n). Specifically, it wants to find the total sum when n takes on the integer values from 2 to 5, inclusive. This means you would calculate four separate terms—4(2), 4(3), 4(4), and 4(5)—and then add those results together to get the final sum.
$\sum_{n = 2}^{5} 4 n$
Answer
Solution:
Step 1:
Write out the terms of the summation for each integer value of $n$ from $2$ to $5$.
$4 \cdot 2 + 4 \cdot 3 + 4 \cdot 4 + 4 \cdot 5$
Step 2:
Perform the simplification process.
Step 2.1:
Calculate $4$ times $2$.
$8 + 4 \cdot 3 + 4 \cdot 4 + 4 \cdot 5$
Step 2.2:
Calculate $4$ times $3$.
$8 + 12 + 4 \cdot 4 + 4 \cdot 5$
Step 2.3:
Combine $8$ and $12$.
$20 + 4 \cdot 4 + 4 \cdot 5$
Step 2.4:
Calculate $4$ times $4$.
$20 + 16 + 4 \cdot 5$
Step 2.5:
Combine $20$ and $16$.
$36 + 4 \cdot 5$
Step 2.6:
Calculate $4$ times $5$.
$36 + 20$
Step 2.7:
Combine $36$ and $20$.
$56$
Knowledge Notes:
The problem at hand involves evaluating a finite arithmetic series. The series is defined by the summation of terms $4n$ where $n$ ranges from $2$ to $5$. The steps taken to solve this problem are as follows:
Expansion of the Series: The series is expanded by substituting the values of $n$ into the expression $4n$ and writing out each term.
Simplification: The terms of the series are simplified by performing the multiplication and addition operations in a step-by-step manner.
Multiplication: Each term $4n$ is simplified by multiplying the constant $4$ with the respective value of $n$.
Addition: The results of the multiplication are then added together in a sequential process to obtain the final sum.
Arithmetic Series: An arithmetic series is a sequence of numbers in which each term after the first is obtained by adding a constant, called the common difference, to the previous term. In this case, the common difference is not directly used, but the series is still arithmetic because it is a linear function of $n$.
Sigma Notation: The summation is expressed using sigma notation, which compactly represents the sum of a sequence of terms. The notation includes the summation symbol ($\Sigma$), the index of summation ($n$ in this case), the lower and upper bounds of summation ($2$ and $5$, respectively), and the general term of the series ($4n$).
Mathematical Operations: Basic mathematical operations such as multiplication and addition are used to evaluate the series.
Final Result: The final result is the sum of the terms of the series when $n$ takes on each integer value from the lower bound to the upper bound.
In this problem, the final result of the summation is $56$.
