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Chapter 11: Problem 47

Verify the formula $$\sum_{x=1}^{n} x=\frac{n(n+1)}{2}$$ for \(n=10,50,\) and \(100.\)

Short Answer

Expert verified
The formula \(\text{LHS} = \text{RHS}\) is verified for \(n=10, 50, \text{ and } 100\) as \(\text{LHS} = \frac{n(n+1)}{2}\).

Step by step solution

01

- Verify for n=10

First, calculate the left-hand side, \(\text{LHS} = \sum_{x=1}^{10} x = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55\). Next, calculate the right-hand side, \(\text{RHS} = \frac{10(10+1)}{2} = \frac{10 \times 11}{2} = 55\). Since \(\text{LHS} = \text{RHS}\), the formula is verified for \(n=10\).
02

- Verify for n=50

First, calculate the left-hand side, \(\text{LHS} = \sum_{x=1}^{50} x = 1 + 2 + 3 + ... + 50 = 1275\). Next, calculate the right-hand side, \(\text{RHS} = \frac{50(50+1)}{2} = \frac{50 \times 51}{2} = 1275\). Since \(\text{LHS} = \text{RHS}\), the formula is verified for \(n=50\).
03

- Verify for n=100

First, calculate the left-hand side, \(\text{LHS} = \sum_{x=1}^{100} x = 1 + 2 + 3 + ... + 100 = 5050\). Next, calculate the right-hand side, \(\text{RHS} = \frac{100(100+1)}{2} = \frac{100 \times 101}{2} = 5050\). Since \(\text{LHS} = \text{RHS}\), the formula is verified for \(n=100\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Arithmetic Series
An arithmetic series is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the 'common difference'. For example, in the series 1, 2, 3, 4, ... each term increases by 1, making the common difference 1. Another example is 2, 5, 8, 11, ... where the common difference is 3.

Arithmetic series can be visualized as a staircase where each step is equal in height. Understanding the concept of arithmetic series is crucial for working with summation formulas, as the sum of the series is our main interest.
Summation Formula
The summation formula for the first n natural numbers is given by:
\[ \sum_{x=1}^{n} x=\frac{n(n+1)}{2} \]
This formula helps us find the sum of the first n terms of an arithmetic series easily, without having to add each term individually. For instance, to find the sum from 1 to 10, you just plug 10 into the formula:
\[ \frac{10(10+1)}{2} = \frac{10 \cdot 11}{2} = 55 \]
This simplifies calculations significantly, especially for large values of n.
Mathematical Verification
Mathematical verification involves proving mathematically that a formula or equation is correct. In verifying the summation formula:
1. First, we calculate the sum of the series manually by adding each term.
2. Next, we compute the sum using the given formula.
3. If the results from steps 1 and 2 are equal, the formula is verified.
For example, for n=10, the sum of the first 10 numbers is:
\[ 1 + 2 + 3 + ... + 10 = 55 \]
Using the summation formula:
\[ \frac{10(10+1)}{2} = 55 \]
Since both results match, the formula is verified for n=10. Repeating this process for n=50 and n=100 will likewise show that the formula is consistently accurate for any value of n.

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Most popular questions from this chapter

Determine the sums of the following geometric series when they are convergent. $$6-1.2+.24-.048+.0096-\cdots$$

Determine the sums of the following infinite series: $$\sum_{k=0}^{\infty} \frac{7}{10^{k}}$$

Find the Taylor series at \(x=0\) of the given function by computing three or four derivatives and using the definition of the Taylor series. $$\frac{1}{2 x+3}$$

Sum an appropriate infinite series to find the rational number whose decimal expansion is given. $$.2727 \overline{27}$$

Use the integral test to determine if \(\sum_{k=1}^{\infty} \frac{e^{1 / k}}{k^{2}}\) is convergent. Show that the hypotheses of the integral test are satisfied.
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