Question

Write out the sums. (You do not need to evaluate them.) \(\sum_{i=-2}^{2}(i+1)^{2}\)

Short answer

Question: Write out the given summation without evaluating the sum: \(\sum_{i=-2}^{2}(i+1)^2\) Answer: The written-out version of the given summation is \(1+0+1+4+9\).

Step-by-step solution

Write down the expression for the given summation

The expression for the summation is given as \(\sum_{i=-2}^{2}(i+1)^2\).

Understand the summation's range

The index \(i\) ranges from \(-2\) to \(2\) with integer steps, which means we will have the following values of \(i\): \(-2, -1, 0, 1, 2\).

Write down each term of the summation

Now we replace the index \(i\) with each of the given values and calculate the corresponding terms. When written as a sum, we will have: \((i+1)^2\bigg|_{i=-2}+(i+1)^2\bigg|_{i=-1}+(i+1)^2\bigg|_{i=0}+(i+1)^2\bigg|_{i=1}+(i+1)^2\bigg|_{i=2}\)

Replace the index \(i\) with its corresponding values

We finally replace \(i\) with the corresponding values in each term to obtain the final sum: \((-2+1)^2+(-1+1)^2+(0+1)^2+(1+1)^2+(2+1)^2\)

Simplify the sum

The sum in a simplified form is: \((-1)^2+(0)^2+(1)^2+(2)^2+(3)^2\) So the written-out version of the summation is: \(1+0+1+4+9\)

Key concepts

Index Range

In the context of summation notation, the index range is crucial for understanding what values we need to work with. Think of the index as a small helper that tells us how many times the expression inside the summation should be evaluated.
In our example, represented by the summation \( \sum_{i=-2}^{2}(i+1)^{2} \), the index \( i \) begins at \( -2 \) and goes up to \( 2 \).

This means the index \( i \) will take each of these integer values:

• -2
• -1
• 0
• 1
• 2

The process of writing out each term becomes clearer once you understand that all integers in this range should be substituted into your expression. Each step inside this range represents one specific calculation. Hence, understanding the index range is the first step in making sense of the complete expression.

Evaluate Expressions

Evaluating expressions is about substituting the values from the index range into the given algebraic expression and calculating the result for each substitution.
For our summation, \((i+1)^{2}\), this expression will be evaluated for every value of \( i \) within our defined range from \(-2\) to \(2\).

Let's break it down:

• When \( i = -2 \), substitute to get \((-2+1)^{2} = (-1)^{2}\) = 1
• When \( i = -1 \), substitute to get \((-1+1)^{2} = (0)^{2}\) = 0
• When \( i = 0 \), substitute to get \((0+1)^{2} = (1)^{2}\) = 1
• When \( i = 1 \), substitute to get \((1+1)^{2} = (2)^{2}\) = 4
• When \( i = 2 \), substitute to get \((2+1)^{2} = (3)^{2}\) = 9

After evaluating the expression for each value of \( i \), we end up with a series of numerical results that, when added together, form the expanded and evaluated expression. Recognizing and carrying out these substitutions is a key technique in simplifying summations.

Algebraic Expressions

Algebraic expressions in the summation are formulas that involve numbers, variables, and arithmetic operations. In our problem, \((i+1)^2\), we have an algebraic expression that involves a variable \( i \), which will change according to the index range.

Each time you substitute \( i \) with a specific value from the range, the resulting expression is purely numerical, and you can simplify it with basic arithmetic.
Here’s why understanding the structure of an algebraic expression is important:

• The operand \((i+1)\) shows us that, initially, each \( i \) is increased by 1 before squaring.
• The exponent forces each result to be positive, reflecting the essence of squaring.

It's the combination of components within the algebraic expression that determines how each term of the summation evolves as \( i \) changes. Recognizing how to work through these expressions builds a stronger foundation for solving more complex algebra problems in the future.