Evaluate the Summation sum from k=2 to 6 of 4k+3
The question asks for a computation of a finite summation. Specifically, you need to calculate the sum of a sequence generated by the function 4k+3 for each integer value of k starting from 2 and ending at 6. This involves applying the function to each integer k within the given range, obtaining the results, and then adding up all of these resulting values to get the total sum.
$\sum_{k = 2}^{6} 4 k + 3$
Answer
Solution:
Step 1:
Write out the terms of the summation for each integer value of $k$ from 2 to 6.
$4 \cdot 2 + 3, 4 \cdot 3 + 3, 4 \cdot 4 + 3, 4 \cdot 5 + 3, 4 \cdot 6 + 3$
Step 2:
Calculate the sum of these terms.
$8 + 3 + 12 + 3 + 16 + 3 + 20 + 3 + 24 + 3 = 95$
Knowledge Notes:
The problem involves evaluating a finite summation, which is a mathematical expression that represents the addition of a sequence of numbers, each generated by substituting the values of an index variable into a given formula. In this case, the index variable is $k$, and the formula is $4k + 3$. The summation is taken from $k=2$ to $k=6$.
To solve this, we follow these steps:
Expansion of the Series: We substitute each integer value of $k$ within the given range into the formula to generate the terms of the series. This step is crucial because it lays out all the individual components that will be summed.
Simplification: After expanding the series, we add up all the terms to find the total sum. This step requires basic arithmetic.
Relevant knowledge points include:
Understanding of summation notation ($\Sigma$), which is a compact way to represent the sum of a sequence of numbers.
Familiarity with arithmetic operations, especially addition.
Recognizing the pattern in the formula provided for the summation, which in this case is linear ($4k + 3$), meaning that as $k$ increases, the terms increase in a linear fashion.
Knowing that the summation of a sequence of numbers is the process of adding them together to find their total.
