Problem 35 - Algebra and Trigonometry with Analytic Geometry

Question

\(\sum_{k=1}^{4}\left(k^{2}-5\right) \quad \)

Step-by-Step Solution

Answer

The sum is 10.

Step 1: Understand the Problem

We need to find the sum of the expression \(k^2 - 5\) for values of \(k\) ranging from 1 to 4. This involves substituting each integer value from 1 to 4 into the expression, calculating the result, and summing all these results.

Step 2: Substitute and Calculate for Each Value of k

Substitute each value of \(k\) into the expression \(k^2 - 5\):- For \(k = 1\): \(1^2 - 5 = 1 - 5 = -4\)- For \(k = 2\): \(2^2 - 5 = 4 - 5 = -1\)- For \(k = 3\): \(3^2 - 5 = 9 - 5 = 4\)- For \(k = 4\): \(4^2 - 5 = 16 - 5 = 11\)

Step 3: Sum All Results

Now, add all the individual results obtained from each value of \(k\):\[-4 + (-1) + 4 + 11\]

Step 4: Calculation of the Final Sum

Calculate the total sum:\[-4 + (-1) = -5\]\[-5 + 4 = -1\]\[-1 + 11 = 10\]

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They are like sentences that use mathematical symbols instead of words. In the exercise, the expression is given as \(k^2 - 5\). This expression contains the variable \(k\), an exponent (squared), and constants. The 2 in \(k^2\) is an exponent that means to multiply the number by itself once (like \(3^2 = 3 \times 3\)). The constant in this expression is \(-5\), which subtracts 5 from the value of \(k^2\).

Generally, algebraic expressions are useful for representing general relationships and solving problems involving unknowns. They help us simplify complex ideas into simpler mathematical terms.

• Variables: Represent unknown values (e.g., \(k\)).
• Exponents: Indicate power to which a number is raised \((k^2\)).
• Constants: Fixed values like \(-5\).

To evaluate an algebraic expression, you replace the variable with a number and perform the operations indicated.

Sum of Squares

The sum of squares is a term used to describe adding up values that have been each squared. In mathematics, squaring a number means to multiply it by itself.

In the given problem, the expression \(k^2 - 5\) involves squaring the variable \(k\). Here, you first evaluate the square of each \(k\) value, from 1 to 4, individually. This means:

• For \(k = 1\), \(k^2 = 1^2 = 1\)
• For \(k = 2\), \(k^2 = 2^2 = 4\)
• For \(k = 3\), \(k^2 = 3^2 = 9\)
• For \(k = 4\), \(k^2 = 4^2 = 16\)

After squaring, you adjust each result by subtracting 5 as per the expression \(k^2 - 5\).
This step-by-step calculation is important when you deal with sequences and series in mathematics, helping ensure accuracy and consistency.

Mathematical Calculation

Mathematical calculation is the process of computing values from given expressions or numbers. It involves steps such as substitution, performing operations, and finding results. Calculations are essential in understanding how numbers interact through operations and in solving complex mathematical problems.
In the step-by-step solution of this exercise, you substitute the numbers 1 through 4 into the expression \(k^2 - 5\), calculate each result and sum them up altogether. Here's how the calculation proceeds:

• Substitute each \(k\) and calculate \(k^2 - 5\):
• For \(k = 1\): \(-4\)
• For \(k = 2\): \(-1\)
• For \(k = 3\): \(4\)
• For \(k = 4\): \(11\)
• Add these results together: \(-4 + (-1) + 4 + 11 = 10\)

Each step needs to be precise to ensure the correct final result. Mathematical calculations help in unraveling the complexities of formulas and offer a clear way to reach solutions.