Problem 116
Question
Exercises \(116-118\) will help you prepare for the material covered in the next section. In Exercises \(116-117\) 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
Verified Answer
The formula \(\frac{n(n+1)}{2}\) is correct for \(n=3\) as both the right and left sides of the equation equate to 6.
1Step 1: Write Down the Given Numbers
We are given that \(n=3\). We can substitute this value into the equation to verify if the given formula is true for \(n=3\).
2Step 2: Apply the Formula
If we plug \(n=3\) into the formula \(\frac{n(n+1)}{2}\), we get \(\frac{3(3+1)}{2} = \frac{3*(4)}{2}\). After simplifying this, we will have \(\frac{12}{2} = 6\).
3Step 3: Calculate the Sum of Numbers
Now calculate the left side of the equation to confirm that the sum of the numbers from 1 to 3 is equal to 6. So, 1 + 2 + 3 = 6.
4Step 4: Compare Both Results
As we can see, both the left side and right side of the equation equals 6. Therefore, we have confirmed that the formula \(\frac{n(n+1)}{2}\) is indeed correct for \(n=3\).
Key Concepts
Arithmetic SeriesSummation FormulaInductive ReasoningVerification for Specific Values
Arithmetic Series
An arithmetic series is the sum of the terms of an arithmetic sequence. In an arithmetic sequence, the difference between consecutive terms is constant. For example, the sequence 1, 2, 3 is an arithmetic sequence because the difference between each term is 1.
Arithmetic sequences are characterized by their first term and the common difference. These characteristics make them simple and predictable.
When we talk about an arithmetic series, we're looking at adding up the terms in an arithmetic sequence. This property allows us to use formulas to calculate sums quickly, even with many terms.
Arithmetic sequences are characterized by their first term and the common difference. These characteristics make them simple and predictable.
When we talk about an arithmetic series, we're looking at adding up the terms in an arithmetic sequence. This property allows us to use formulas to calculate sums quickly, even with many terms.
Summation Formula
The summation formula for an arithmetic series greatly simplifies the task of finding the sum of a large number of terms. The general formula is \( S_n = \frac{n(n+1)}{2} \), where \(n\) is the number of terms.
This formula was famously derived by the mathematician Carl Friedrich Gauss when he was a schoolboy. It reduces the workload significantly because instead of adding each number individually, you use the formula to find the sum instantly.
Say you want to verify the formula for \( n = 3 \). Substituting \( n = 3 \) into the formula, we get \( \frac{3(3+1)}{2} = 6 \), which matches the manual sum 1 + 2 + 3.
This formula was famously derived by the mathematician Carl Friedrich Gauss when he was a schoolboy. It reduces the workload significantly because instead of adding each number individually, you use the formula to find the sum instantly.
Say you want to verify the formula for \( n = 3 \). Substituting \( n = 3 \) into the formula, we get \( \frac{3(3+1)}{2} = 6 \), which matches the manual sum 1 + 2 + 3.
Inductive Reasoning
Inductive reasoning is an important concept in mathematics used to prove that a statement or formula is true for all natural numbers. It involves two main steps:
In the case of our arithmetic series, once you establish the base case and prove the pattern holds, you can confidently use the formula for any number of terms.
- Base Case: Verify the statement is true for the initial value, usually \( n = 1 \).
- Inductive Step: Assume the statement is true for some \( n = k \), and then show it must also be true for \( n = k+1 \).
In the case of our arithmetic series, once you establish the base case and prove the pattern holds, you can confidently use the formula for any number of terms.
Verification for Specific Values
Verification for specific values involves checking if a formula works by substituting particular numbers into it. This serves as a practical check rather than a theoretical proof.
For example, with our formula \( S_n = \frac{n(n+1)}{2} \), if you want to verify it for \( n = 3 \), you substitute \( n = 3 \) into both the formula and manual addition to compare results.
This method is crucial when first learning about new formulas because it builds intuition about why a formula might be true. Although it doesn't replace a full inductive proof, it provides a good confidence boost that the formula works for at least some cases. This practice is especially helpful in exercises where you prove formulas for multiple values.
For example, with our formula \( S_n = \frac{n(n+1)}{2} \), if you want to verify it for \( n = 3 \), you substitute \( n = 3 \) into both the formula and manual addition to compare results.
This method is crucial when first learning about new formulas because it builds intuition about why a formula might be true. Although it doesn't replace a full inductive proof, it provides a good confidence boost that the formula works for at least some cases. This practice is especially helpful in exercises where you prove formulas for multiple values.
Other exercises in this chapter
Problem 114
Convert the equation $$ 4 x^{2}+y^{2}-24 x+6 y+9=0 $$ to standard form by completing the square on \(x\) and \(y .\) Then graph the ellipse and give the locatio
View solution Problem 115
Use a right triangle to write \(\cos \left(\tan ^{-1} x\right)\) as an algebraic expression. Assume that \(x\) is positive and that the given inverse trigonomet
View solution Problem 117
Exercises \(116-118\) will help you prepare for the material covered in the next section. In Exercises \(116-117\) show that $$1+2+3+\cdots+n=\frac{n(n+1)}{2}$$
View solution Problem 118
Exercises \(116-118\) will help you prepare for the material covered in the next section. In Exercises \(116-117\) show that $$1+2+3+\cdots+n=\frac{n(n+1)}{2}$$
View solution