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Series expansion is the process of writing out each term of a summation to make it easier to work with. In this case, we are given a summation expression: \( \sum_{k=1}^{6}(k^2 - 5) \). This instructs us to evaluate the expression \( k^2 - 5 \) for every integer value \( k \) from 1 to 6.
The series begins with the smallest value of \( k \), which is 1, and increases step-by-step up to 6. At each step, the expression \( k^2 - 5 \) is evaluated. By writing out each term explicitly, one can see the individual contributions to the overall sum.

• For \( k = 1 \), the expression becomes \( 1^2 - 5 = -4 \).
• For \( k = 2 \), the calculation yields \( 2^2 - 5 = -1 \).
• Continuing in this way, the results are \( 4, 11, 20, \) and \( 31 \) for \( k = 3, 4, 5, 6 \) respectively.

By laying out each step clearly, series expansion helps in simplifying calculations and allows you to organize complex expressions into manageable parts.

Once you have expanded the series and worked out each term, the next step is arithmetic calculation. This means adding up all those values you calculated in the series expansion.
The process involves basic arithmetic operations (in this case, addition) that are performed sequentially to arrive at a final sum. Let's review this calculation:
We have the following terms from our expansion:

• \(-4\)
• \(-1\)
• \(4\)
• \(11\)
• \(20\)
• \(31\)

Add these step-by-step:

• Start by adding \(-4 + (-1)\) to get \(-5\).
• Then add \(-5 + 4\) to reach \(-1\).
• Next, calculate \(-1 + 11\) resulting in \(10\).
• Proceed with \(10 + 20\) which gives \(30\).
• Finally, adding \(30 + 31\) you arrive at \(61\).

Each calculation involves simple addition, staying organized makes it easier to avoid mistakes, ensuring the correct final sum.

Evaluating algebraic expressions like \( k^2 - 5 \) involves substituting values and performing mathematical operations. Algebraic expressions can include constants, variables, and operations like addition, subtraction, multiplication, or division.

In the given exercise, the expression \( k^2 - 5 \) means:

• \( k^2 \) is the square of the variable \( k \).
• Subtract \( 5 \) from \( k^2 \).

When you substitute values for \( k \) (specifically, 1 through 6 in this case), you are solving the expression for each of those values. Calculating \( k^2 \) means multiplying \( k \) by itself, and the subtraction then adjusts the outcome according to the expression format.
This demonstrates a common task in algebra: plug in different values into an expression to get specific results, which are then summed up if required by the context of the problem, like in our summation. Understanding how to work with algebraic expressions is fundamental for solving many mathematical problems.