Solve the problem involving sum of a series

Therefore, ## \sum_{r=1}^n r=\dfrac{1}{2} \left[n^2+n\right] ##. In summary, the sum of the first n natural numbers is equal to half of n squared plus n. This can be obtained by simplifying the expression r(r+1)-(r-1)r and using the method of difference.
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chwala
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Homework Statement
Simplify ##r(r+1)-(r-1)r## and use your result to obtain ## \sum_{r=1}^n r##
Relevant Equations
Method of difference
1671676295280.png
My attempt;

##r^2+r-r^2+r=2r##

Let ##f(r)=(r-1)r## then it follows that ##f(r+1)=r(r+1)## so that ##2r## is of the form ##f(r+1)-f(r)##.

When

##r=1;## ##[2×1]=2-0##
##r=2;## ##[2×2]=6-2##
##r=3;## ##[2×3]=12-6##
##r=4;## ##[2×4]=20-12##

...

##r=n-1##, We shall have ##2(n-1)=n-1(n)-(n-2)(n-1)##

##r=n##, We shall have ##2n=n(n+1)-(n-1)n## ## 2\sum_{r=1}^n r=n(n+1)-0##

## 2\sum_{r=1}^n r=n^2+n##

## \sum_{r=1}^n r=\dfrac{1}{2} \left[n^2+n\right]##

your insight is welcome...
 
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I mean, do we have to make things complicated here or what?
 
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chwala said:
Homework Statement:: Simplify ##r(r+1)-(r-1)r## and use your result to obtain ## \sum_{r=1}^n r##
Relevant Equations:: Method of difference

Let f(r)=(r−1)r then it follows that f(r+1)=r(r+1) so that 2r is of the form f(r+1)−f(r).
[tex]2r=f(r+1)-f(r)[/tex]
[tex]2\sum_{r=1}^nr=\sum_{r=1}^nf(r+1)-\sum_{r=1}^nf(r)=\sum_{r=1}^nf(r+1)-\sum_{r=0}^{n-1}f(r+1)=f(n+1)-f(1)=n(n+1)[/tex]
 
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1. What is the formula for finding the sum of a series?

The formula for finding the sum of a series is S = n(a1 + an) / 2, where S is the sum, n is the number of terms, and a1 and an are the first and last terms of the series, respectively.

2. How do I know if a series is arithmetic or geometric?

An arithmetic series has a common difference between each term, while a geometric series has a common ratio between each term. To determine which type of series you have, look for a consistent change in values between each term. If the difference is the same, it is an arithmetic series. If the ratio is the same, it is a geometric series.

3. Can I use the sum of a series formula for infinite series?

Yes, the formula for finding the sum of a series can be used for both finite and infinite series. However, for infinite series, the formula may only work if the series converges, meaning that the sum approaches a finite value as the number of terms increases.

4. What is the difference between a series and a sequence?

A sequence is a list of numbers in a specific order, while a series is the sum of those numbers. In other words, a series is the sum of a sequence.

5. How can I apply the sum of a series in real life?

The sum of a series can be applied in various real-life scenarios, such as calculating the total cost of a loan with compound interest, finding the total distance traveled in a trip with varying speeds, or determining the total revenue of a company with increasing sales over time. It can also be used in mathematics and physics to solve problems involving infinite sums, such as the area under a curve or the total energy of a system.

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