Problem 112 - College Algebra

Question

Exercises will help you prepare for the material covered in the next section. In Exercises \(112-113,\) show that $$ 1+2+3+\cdots+n-\frac{n(n+1)}{2} $$is true for the given value of \(n .\) $$n-3: \text { Show that } 1+2+3-\frac{3(3+1)}{2}.$$

Step-by-Step Solution

Answer

The equation \(1+2+3 - \frac{3(3+1)}{2} = 1+2+3-6 = 0\) is proven true for n = 3.

Step 1: Substitute Value

Substitute the given values into the equation. We are given n = 3, so the equation becomes: \(1+2+3 - \frac{3(3+1)}{2}\) . This simplifies to \(1+2+3 - \frac{3(4)}{2} = 1+2+3-6 = 0\)

Step 2: Check Both Sides Of The Equation

Make sure both sides of the equation match. In this case, they do. The left side (1+2+3 - 6) equals 0 as does the right side. Since both sides match, the equation is true for n=3.

Key Concepts

Substitution in Equations

In mathematics, substitution in equations is a key technique used to solve, simplify, or verify equations. It involves replacing a variable in an equation with a specific value or another variable. This method helps in evaluating expressions or proving statements for certain values. In our given exercise, we substitute the specified value, which is \(n = 3\), into the expression \[1 + 2 + 3 + \cdots + n - \frac{n(n+1)}{2}\]. By substituting \(n = 3\) into the equation, we transform it into \[1 + 2 + 3 - \frac{3(3+1)}{2}.\] This substitution allows us to focus on defining this expression for a particular value of \(n\), enabling further simplification and verification.

Algebraic Manipulation

Algebraic manipulation is a foundational skill in problem-solving that involves rearranging expressions and equations using algebraic rules. Here, we aim to simplify both parts of the equation. We start with the substituted expression:- The summation part: \(1+2+3 = 6\).- The formula part: \(\frac{3 \times 4}{2} = 6\).After these calculations, we simplify the full expression, which initially was \[1+2+3-\frac{3(3+1)}{2} = 6 - 6 = 0.\]Breaking down each step, we leverage algebraic rules, like distributing multiplication over addition, to simplify complex parts into manageable pieces. Understanding these steps helps us convert seemingly complicated expressions into straightforward equations.

Checking Solutions

Checking solutions is a crucial step in confirming the validity of an equation. It's a process of verifying whether the solution satisfies the given equation and ensures the calculation's correctness. In this exercise, we have the simplified equation, \[1+2+3-rac{3(4)}{2} = 0.\]To check our solution:

• Ensure both sides of the equation equal after substitution: Here, both sides are 0.
• Verify each arithmetic step within the equation to catch any mistakes in calculation.
• Ensure logical consistency of the manipulations.

When left and right sides match after simplification, we've verified our solution is accurate. This reaffirms that when \(n = 3\), the original expression equals zero.

Summation Formula

The summation formula is a powerful tool used to add sequences of numbers efficiently. Here, the formula used is:\[S_n = \frac{n(n+1)}{2},\]which calculates the sum of all natural numbers from 1 to \(n\). This equation is derived from the properties of arithmetic progressions and is especially useful for quickly summing large ranges of consecutive numbers without enumerating each term.In the exercise, the formula helps us compute the series sum up to 3:- Substitute \(n = 3\) in the formula: \(\frac{3(3+1)}{2} = 6\).- Compare with the manual sum: \(1+2+3 = 6\).Using the summation formula verified the result, illustrating its utility and efficiency.Hence, acknowledging pattern recognition in sequences with such formulas helps in mastering mathematical concepts.