Consecutive sum of exponentiations

  • Thread starter scientifico
  • Start date
  • Tags
    Sum
In summary, the sum of exponentiations in the form of x^0+x^1+x^2+...+x^n can be solved using the formula \frac{x^(n+1)-1}{x-1}. This is because the expansion of (x^0+x^1+x^2+...+x^n)(x-1) can be simplified to \frac{x^(n+1)-1}{x-1}. This can be demonstrated by multiplying the series by (x-1) and simplifying.
  • #1
scientifico
181
0
Hello, i read that a sum of exponentiations like [itex]x^0+x^1+x^2+x^3...+x^n[/itex] can be solved with this forumula [itex]\frac{x^(n+1)-1}{x-1}[/itex], how is it possible do demonstrate this resolutive formula?

Thank you!
 
Mathematics news on Phys.org
  • #2
Simply expand :
(x^0+x^1+x^2+...+x^n)(x-1)
 
  • #3
I didn't understand why you wrote (x-1) after the expansion... thank you
 
  • #4
Because of the x- 1 in the denominator in your equation.
If A= B/(x- 1) then A(x- 1)= B.

Is that true here?
x
That is not, however, how I would handle this problem. I would note that [itex]x^0+ x^1+ x^2+ \cdot\cdot\cdot+ x^n[/itex] is a geometric series with "common factor" x. The sum of a finite geometric series is
[tex]\frac{1- x^{n+1}}{1- x}= \frac{x^{n+1}- 1}{x- 1}[/tex].
 
  • #5
scientifico said:
I didn't understand why you wrote (x-1) after the expansion... thank you

You will understand if you make the multiplication of the series by (x-1) and then simplify. Just do it !
 

1. What is a consecutive sum of exponentiations?

A consecutive sum of exponentiations is a mathematical operation where a number is raised to different powers in a sequence, and then all of the results are added together. For example, the consecutive sum of exponentiations for the number 2 would be 2^1 + 2^2 + 2^3 + ... + 2^n.

2. How is a consecutive sum of exponentiations calculated?

To calculate a consecutive sum of exponentiations, you first need to determine the number you will be raising to different powers (also known as the base). Then, you need to determine the number of times you will raise the base to different powers (also known as the exponent). Finally, you can use a mathematical formula or a calculator to compute the sum of all the individual exponentiations.

3. What is the purpose of calculating a consecutive sum of exponentiations?

Consecutive sums of exponentiations are often used in mathematical and scientific fields to solve complex problems or equations. They can also help with understanding patterns and relationships between numbers.

4. Can a consecutive sum of exponentiations be used with any number?

Yes, a consecutive sum of exponentiations can be used with any number. However, the results may vary depending on the base and the number of exponentiations used. Some numbers may result in a finite sum, while others may result in an infinite sum.

5. Are there any real-life applications for consecutive sums of exponentiations?

Yes, consecutive sums of exponentiations have various real-life applications, such as in finance for calculating compound interest, in physics for calculating energy levels, and in computer science for calculating time complexity of algorithms.

Similar threads

Replies
4
Views
285
  • General Math
Replies
6
Views
1K
  • General Math
Replies
10
Views
809
  • General Math
Replies
6
Views
804
Replies
3
Views
576
  • General Math
Replies
5
Views
1K
Replies
12
Views
900
  • General Math
Replies
1
Views
687
  • General Math
Replies
5
Views
2K
  • General Math
Replies
1
Views
1K
Back
Top