Evaluate the Summation sum from k=-2 to 2 of 2k+1
The problem asks for the evaluation of a mathematical expression in the form of a summation. Specifically, you are required to calculate the sum of the terms generated by the expression $2k+1$as the integer $k$varies from $-2$to $2$. This involves finding the values of the expression when substituting $k$with each integer in the given range and then adding all those values together to get the final sum.
$\sum_{k = - 2}^{2} 2 k + 1$
Answer
Solution:
Step 1: Expansion of the Summation
Write out the terms of the series by substituting the values of $k$ from $-2$ to $2$ into the expression $2k+1$. The series becomes:
$$(2 \cdot -2 + 1) + (2 \cdot -1 + 1) + (2 \cdot 0 + 1) + (2 \cdot 1 + 1) + (2 \cdot 2 + 1)$$
Step 2: Calculation of the Sum
Combine the terms to find the sum of the series:
$$(-4 + 1) + (-2 + 1) + (0 + 1) + (2 + 1) + (4 + 1) = 5$$
Knowledge Notes:
The question involves evaluating a finite summation, which is a process of adding up all the terms in a sequence defined by a formula. The given summation is:
$$\sum_{k=-2}^{2} (2k+1)$$
This notation indicates that we should substitute each integer value of $k$ from $-2$ to $2$ into the formula $2k+1$ and then add up all the resulting values.
Relevant knowledge points include:
Summation Notation: The symbol $\sum$ represents the sum of a sequence of numbers. The variable below the summation symbol is the index of summation, and the numbers below and above the symbol indicate the starting and ending values, respectively.
Arithmetic Sequences: The given formula $2k+1$ generates an arithmetic sequence because the difference between consecutive terms is constant (in this case, the difference is 2).
Evaluating Summations: To evaluate a finite summation, one can either calculate each term individually and then add them up, or use formulas for specific types of sequences, such as arithmetic or geometric sequences.
Simplification: After expanding the series, it is important to simplify the expression by performing the indicated operations (addition, subtraction, multiplication, etc.) to find the final sum.
In this problem, we use the first method, which involves expanding the series and then simplifying to find the sum. The series is expanded by substituting the values of $k$ from $-2$ to $2$ into the expression $2k+1$, and the sum is found by adding the resulting values.
